We prove a local minimizing property for strictly stable free-boundary minimal hypersurfaces in the relative current setting. Let \(\Sigma^n\) be a compact, two-sided, properly embedded free-boundary minimal hypersurface
in a compact Riemannian manifold \((N^{n+1},\partial N)\). If \(\Sigma\) is strictly stable, then, in a sufficiently small free-boundary adapted tubular neighborhood \(K_r\), the relative cycle \(\llbracket\Sigma\rrbracket\) is the unique mass minimizer in its relative \(\mathbb{Z}_2\)-homology class in \((K_r,K_r\cap\partial N)\).
We further prove a relative flat-neighborhood version, and apply this to obtain an index-one conclusion for a multiplicity-one realization of the first free-boundary width under the standard generic hypotheses.
The main point is to bridge the gap between strict stability, which is a smooth graphical condition, and local minimality among relative cycles. We prove that any relative mass minimizer in the same class converges to \(\Sigma\) with multiplicity one as a varifold, satisfies a uniform first variation bound, and hence becomes a small free-boundary graph by Allard–Grüter–Jost regularity. The conclusion then follows from the strict stability of
\(\Sigma\).
A basic principle in finite-dimensional variational theory is that a strictly stable critical point is a strict local minimizer. More precisely, if \(f\) is a smooth function on a Banach space and \(x_0\) is a critical point such that the second variation of \(f\) at \(x_0\) is positive definite, then, after shrinking the neighborhood if necessary, \[f(x)>f(x_0)\qquad \text{for every \(x\neq x_0\) sufficiently close to \(x_0\).}\] In this setting the statement is essentially immediate from Taylor expansion.
However, the analogous question for geometric variational problems is subtler. Suppose that \(\Sigma\) is a strictly stable critical point of the area functional, or more generally of an elliptic parametric functional.
This leads to the following natural question:
Question 1. Does strict stability of a critical submanifold imply that it is a strict local minimizer among all admissible geometric competitors in a sufficiently small neighborhood?
The difficulty in Question 1 lies in the gap between stability and local minimality. Strict stability is detected by the second variation and is therefore a statement about smooth nearby
deformations. However, the natural competitors in geometric measure theory need not be smooth hypersurfaces at all. They may be integral currents, flat chains, or varifolds, and may carry singularities, multiplicity, or non-graphical sheets. Consequently,
one cannot simply apply a Taylor expansion of the area functional near \(\Sigma\). One must first show that any potential minimizer among weak competitors is forced, by its variational properties, to become graph near \(\Sigma\).
A fundamental result of B. White gives a positive answer in the closed and fixed boundary settings. That is:
Theorem 2 ([1]). Let \(M\) be a smooth embedded compact manifold with (possibly empty) boundary and
suppose \(M\) is strictly stable for a smooth parametric elliptic integrand \(F\). Then there is an open set \(U\) containing \(M\) such that \[F(M)\leq (M')\] whenever \(M'\) is a current that is homologous to \([M]\) in \(\bar U.\) Equality holds if and only if \(M'=\llbracket M\rrbracket.\)
This result provides an important bridge between infinitesimal stability and local minimality in the weak topology of geometric measure theory. In the case where \(M\) has boundary, White’s theorem is a fixed-boundary
statement: the competitors have the same boundary as \(M\). Thus White’s theorem does not directly address the free-boundary situation, where the boundary of the competitor is allowed to move along the ambient boundary.
This leads to the following free-boundary analogue of question 1:
Question 3. Is a strictly stable free-boundary minimal submanifold the unique local minimizer in its relative homology class among all nearby relative cycles?
The main result of this paper gives an affirmative answer to question 3 in the codimension-one case. We prove that strict stability forces local mass-minimality not only among smooth
free-boundary graphs, but among all nearby relative \(\mathbb{Z}_2\)-cycles in the same relative homology class. That is:
Theorem 4 (Main Theorem). Let \((N^{n+1},g)\) be a compact smooth Riemannian manifold with boundary. Let \(\Sigma^n\subset N\) be a compact, two-sided, properly embedded
free-boundary minimal hypersurface. Assume that \(\Sigma\) is strictly stable for area functional, then there exist \(r_0>0\) and a free-boundary adapted tubular parametrization \[F:\Sigma\times (-r_0,r_0)\to N\] onto its image, satisfying \[F(x,0)=x,
\qquad
F(\partial\Sigma\times (-r_0,r_0))\subset \partial N,\] such that the following holds: for \(0<r<r_0\), set \[K_r:=F(\Sigma\times[-r,r]),
\qquad
A_r:=K_r\cap \partial N.\] If \(T\in \mathcal{Z}_n(K_r,A_r;\mathbb{Z}_2)\) satisfies \[[T]=[\Sigma]\in H_n(K_r,A_r;\mathbb{Z}_2),\] then \[\mathbf{M}(T)\ge
\mathbf{M}(\Sigma).\] Moreover, equality holds if and only if \(T=\llbracket\Sigma\rrbracket\) as a relative \(\mathbb{Z}_2\)-cycle in \((K_r,A_r)\).
We also prove a relative flat-neighborhood version of the theorem. The tubular-neighborhood formulation is well adapted to the proof, but min–max sweepouts are naturally controlled in the flat or \(\mathbf{F}\)-topology,
and flat closeness does not imply support containment in a tubular neighborhood. Theorem 5 removes this mismatch: it shows that strict stability gives local minimality in an actual
relative flat neighborhood of \(\llbracket\Sigma\rrbracket\). This gives the following flat-neighborhood form of the local minimizing property:
Theorem 5 (Relative flat-neighborhood local minimality). With all assumptions in Theorem 4, there exists \(\varepsilon>0\) such that
the following holds:
If \[S\in\mathcal{Z}_n(N,\partial N;\mathbb{Z}_2),
\qquad
[S]=[\llbracket\Sigma\rrbracket]\in H_n(N,\partial N;\mathbb{Z}_2),\] and \[\mathcal{F}_{\mathrm{rel}}^N(S-\llbracket\Sigma\rrbracket)<\varepsilon,\] then \[\mathbf{M}(S)\ge
\mathbf{M}(\llbracket\Sigma\rrbracket).\] Moreover, if equality holds, then \(S=\llbracket\Sigma\rrbracket\) as a relative \(\mathbb{Z}_2\)-cycle in \((N,\partial N)\).
As an application, under the standard generic hypotheses in the free-boundary min–max theory, we show that a multiplicity-one realization of the first free-boundary width has total Morse index one. More precisely, using the multiplicity-one theorem of
Sun–Wang–Zhou [2] and the Morse index upper bound of Guang–Li–Wang–Zhou [3], the flat-neighborhood local minimality theorem rules out the possibility that this realization is strictly stable.
1.2 Related Works and Motivation from Min-Max Theory↩︎
The result most directly related to the present paper is White’s local minimality theorem for strictly stable critical submanifolds [1]. White proved
that if a smooth compact embedded submanifold is stationary and strictly stable for a smooth parametric elliptic functional, then it is minimizing, in a sufficiently small neighborhood, among all currents in the same homology class. In the case with
boundary, this is a fixed-boundary statement: the competitors are required to have the same boundary.
White’s theorem is part of a broader local Morse-theoretic picture. In the same paper, White also treated nondegenerate critical submanifolds of positive index [1]. More precisely, if a critical submanifold has index \(k\) and nullity zero, then it admits a local \(k\)-parameter min–max characterization. Thus the
strictly stable case may be viewed as the index-zero case of a local min–max theory for geometric variational problems. This local Morse-theoretic viewpoint is closely related to the deformation arguments used in later works on the Morse index of min–max
hypersurfaces, see for example [4].
Several refinements and variants of this local minimality principle are known. Inauen–Marchese proved a quantitative version of White’s strictly stable case, where the tubular neighborhood is replaced by a flat-distance neighborhood and one obtains a
quantitative lower bound for the excess energy [5]. Morgan–Ros proved an \(L^1\)-neighborhood local minimality theorem for
strictly stable constant-mean-curvature hypersurfaces under a fixed-volume constraint [6]. Federer’s earlier work also contains related local homological
minimality results for compact minimal submanifolds with boundary [7]. Related ideas also appear in quantitative isoperimetric stability and nonlocal
isoperimetric problems, where selection principles and second-variation arguments are used to obtain weak local minimality or quantitative stability [8]–[10].
The present paper fits into this local theory, but in the free-boundary setting. The main difference from the fixed-boundary situation is that the boundary of a competitor is not prescribed; it is allowed to move along the ambient boundary. Thus the
natural class of competitors is not a fixed-boundary current class, but a relative \(\mathbb{Z}_2\)-cycle class. Our main theorem proves the corresponding local minimality statement for strictly stable properly embedded
free-boundary minimal hypersurfaces in codimension one.
Free-boundary minimal hypersurfaces have also been studied extensively from the viewpoint of regularity, compactness, and stability. The free-boundary varifold regularity theory of Grüter–Jost provides the boundary analogue of Allard’s regularity
theorem [11], [12]. Compactness and graphical convergence results were obtained
by Fraser–Li and by Ambrozio–Carlotto–Sharp, while Guang–Zhou proved compactness and generic finiteness results under area and index bounds [13]–[15].
We finally mention the min–max motivation. The Almgren–Pitts theory, originating in the work of Almgren and Pitts and relying on the regularity theory of Schoen–Simon, has been highly successful in producing minimal hypersurfaces and in organizing them
according to variational complexity, such as widths, multiplicity, and Morse index [4], [16]–[20]. For min–max constructions with fixed boundary, see De Lellis–Ramic [21]; this should be distinguished from the relative-cycle framework used in the free-boundary setting, where the boundary of a competitor is allowed to move along the ambient boundary. For this setting, the Almgren–Pitts
theory was developed by Li–Zhou, general Morse index upper bounds were proved by Guang–Li–Wang–Zhou, and the multiplicity-one theorem for generic metrics was proved by Sun–Wang–Zhou [2], [3], [22]. Since free-boundary min–max sweepouts are naturally formulated in spaces of relative cycles, the
local relative-current theorem proved here is expected to be useful for the corresponding index problem. In particular, the flat-neighborhood version proved in Section 4 gives a local input toward the index-one property for multiplicity-one realizations of
the first free-boundary width.
We briefly explain the strategy of the proof of Theorem 4. The main point is that strict stability is directly useful only after the competitor has been shown to be a free-boundary
graph over \(\Sigma\). Thus the proof is organized as a reduction from arbitrary relative cycles to graphical competitors.
Although the result is motivated by White’s local minimality theorem, our proof uses a different approach: we avoid the almost-minimizing framework and instead proceed through projection estimates, multiplicity-one varifold convergence, a uniform first
variation bound, and then apply the interior and free-boundary Allard regularity theorems.
A natural alternative would be to double \(N\) across \(\partial N\) and try to apply a closed local minimality theorem. This reduction is not automatic: the doubled metric may fail to be
smooth, and free-boundary strict stability, which is governed by a Robin boundary problem, does not directly imply strict stability of the doubled hypersurface for all variations. We therefore work directly in the relative-current setting.
The proof has three main ingredients. First, we use the tubular projection onto \(\Sigma\) to compare the mass of an arbitrary relative minimizer with the mass of \(\Sigma\). The relative
homology condition, together with the constancy theorem, shows that the projection of such a minimizer represents exactly the fundamental relative class of \(\Sigma\). A Jacobian estimate for the tubular projection then
implies that any minimizing sequence in the relative class must have mass converging to the mass of \(\Sigma\), and hence converges to \(\Sigma\) as a varifold with multiplicity one.
Second, we prove a uniform first variation bound for these minimizers. Since the minimization is performed inside a tubular neighborhood, an arbitrary free-boundary variation may leave the neighborhood. To deal with this, we compose the variation with a
Lipschitz retraction back to the tubular neighborhood. This produces admissible competitors in the same relative homology class and gives a first variation estimate independent of the size of the neighborhood.
Third, we apply regularity theory. At interior points we use Allard’s regularity theorem [[11]][23], while at boundary points we use the free-boundary regularity theorem of Grüter–Jost [12]. The multiplicity-one convergence and the uniform first variation bound imply that, for sufficiently small tubular neighborhoods, every relative mass minimizer is in fact a free-boundary graph over \(\Sigma\). At this point strict stability applies: any nontrivial sufficiently small graph over \(\Sigma\) has strictly larger area. Therefore the only possible minimizer is \(\llbracket\Sigma\rrbracket\) itself.
The flat-neighborhood version requires one additional idea. Relative flat closeness to \(\Sigma\) does not force the support of a competitor to lie in the tubular neighborhood \(K_r\).
Following the exterior-replacement strategy of Marques–Neves [4], we fix a smaller neighborhood \(V\Subset K_r\) and minimize
mass among relative cycles which agree with the given competitor inside \(V\). The resulting exterior replacement is stationary with free boundary in \(N\setminus\overline{V}\). The
free-boundary monotonicity formula then implies that if its support reached \(N\setminus K_r\), it would carry a definite amount of mass outside \(V\), contradicting flat convergence to
\(\Sigma\). Hence the replacement is forced into \(K_r\), where Theorem 4 applies.
Finally, in the first-width application, the flat-neighborhood theorem is used as an avoidance result. If a multiplicity-one critical cycle realizing \(\omega_1(N,g)\) had index zero, then under the bumpy metric
assumption it would be strictly stable. Its flat local minimizing neighborhood would then be avoidable by any sufficiently good nontrivial one-sweepout, contradicting the fact that the cycle occurs as a min–max critical element. Together with the
free-boundary index upper bound, this gives total index one.
In Section 2, we collect the basic notation and preliminary facts concerning relative \(\mathbb{Z}_2\)-cycles, varifolds, free-boundary variations, and strict stability.
In Section 3, we prove the main theorem. More precisely, in Section 3.1 we construct a free-boundary adapted tubular neighborhood of \(\Sigma\). In Section 3.2 we prove current and varifold convergence of the relative mass minimizers to \(\llbracket\Sigma\rrbracket\) and \(|\Sigma|\), respectively. In Section 3.3 we establish a uniform first variation bound. In Section 3.4 we apply Allard’s interior regularity theorem and the free-boundary regularity theorem of Grüter–Jost to obtain graphical convergence.
Finally, in Section 3.5 we use strict stability to prove the local minimizing property and complete the proof of the main theorem.
In Section 4, we prove a relative flat-neighborhood version of the local minimizing theorem by an exterior-replacement argument. We then apply this version to the first free-boundary width and prove an index-one conclusion for a multiplicity-one
realization under the standard generic hypotheses.
The authors are supported by the National Natural Science Foundation of China (Grant no. 12371055). The second author would like to thank Wenduo Zou for helpful discussions related to this work.
We first recall some notation for relative currents and relative cycles. These are standard in geometric measure theory and in the free-boundary min–max literature with the notation slightly modified for brevity. See for example, Federer [24], Simon [23] and Guang–Li–Wang–Zhou [3]. Throughout this paper, currents are taken with coefficients in \(\mathbb{Z}_2\), thus signs are irrelevant.
Let \((K,A)\) be a compact pair. We denote by \(\mathcal{R}_k(K;\mathbb{Z}_2)\) the space of \(k\)-dimensional rectifiable currents modulo \(2\) supported in \(K\). Define \[Z_k(K,A;\mathbb{Z}_2)
:=
\left\{
T\in \mathcal{R}_k(K;\mathbb{Z}_2):
\operatorname{spt}(\partial T)\subset A
\right\}.\] Thus an element of \(Z_k(K,A;\mathbb{Z}_2)\) is a current whose boundary is allowed to lie in \(A\).
We say that two elements \(T,S\in Z_k(K,A;\mathbb{Z}_2)\) are equivalent if \(T-S\in \mathcal{R}_k(A;\mathbb{Z}_2),\) that is, if they differ by a current supported in \(A\). We denote the quotient space by \[\mathcal{Z}_k(K,A;\mathbb{Z}_2)
:=
Z_k(K,A;\mathbb{Z}_2)/\mathcal{R}_k(A;\mathbb{Z}_2).\] Elements of \(\mathcal{Z}_k(K,A;\mathbb{Z}_2)\) are called relative \(k\)-cycles in the pair \((K,A)\).
For \(\tau\in \mathcal{Z}_k(K,A;\mathbb{Z}_2),\) we denote by \([\tau]\in H_k(K,A;\mathbb{Z}_2)\) its relative homology class. Equivalently, if \(T,S\in
Z_k(K,A;\mathbb{Z}_2)\) are representatives, then \[[T]=[S]\in H_k(K,A;\mathbb{Z}_2)\] if and only if there exist \[Q\in \mathcal{R}_{k+1}(K;\mathbb{Z}_2),
\qquad
R\in \mathcal{R}_k(A;\mathbb{Z}_2),\] such that \[T-S=\partial Q+R.\]
For every \(\tau\in\mathcal{Z}_k(K,A;\mathbb{Z}_2),\) there is a canonical representative \(T\in\tau\) satisfying \(T\llcorner A=0.\) We shall often
identify a relative cycle with its canonical representative. With this convention, if \(\Sigma^k\subset K\) is a properly embedded smooth submanifold with \(\partial\Sigma\subset A,\) then
\[\llbracket\Sigma\rrbracket\in \mathcal{Z}_k(K,A;\mathbb{Z}_2)\] denotes the relative cycle induced by \(\Sigma\). We also write \([\Sigma]\) for the
relative homology class \([\llbracket\Sigma\rrbracket]\in H_k(K,A;\mathbb{Z}_2).\)
The mass of a relative cycle \(\tau\in\mathcal{Z}_k(K,A;\mathbb{Z}_2)\) is defined by \[\mathbf{M}(\tau)
:=
\inf\{\mathbf{M}(T):T\in\tau\}.\] Equivalently, if \(T\) is the canonical representative of \(\tau\), then \(\mathbf{M}(\tau)=\mathbf{M}(T).\)
We shall also use the relative flat norm. For \(S\in \mathcal{R}_k(K;\mathbb{Z}_2),\) define \[\mathcal{F}_{\mathrm{rel}}(S)
:=
\inf
\left\{
\mathbf{M}(P)+\mathbf{M}(Q):
S=P+\partial Q+R
\right\},\] where the infimum is taken over \[P\in \mathcal{R}_k(K;\mathbb{Z}_2),
\qquad
Q\in \mathcal{R}_{k+1}(K;\mathbb{Z}_2),
\qquad
R\in \mathcal{R}_k(A;\mathbb{Z}_2).\] Thus, in the relative flat norm, an error supported in \(A\) is ignored. Equivalently, for two relative cycles represented by \(T_1,T_2\), one
may write \[\mathcal{F}_{\mathrm{rel}}(T_1-T_2)
=
\inf_{R\in\mathcal{R}_k(A;\mathbb{Z}_2)}
\mathcal{F}(T_1-T_2+R).\] This is the quotient flat norm on the relative cycle space \(\mathcal{Z}_k(K,A;\mathbb{Z}_2)\).
We shall also use the homotopy formula for currents, see [24]. Let \(H:[0,1]\times K\to K'\) be a Lipschitz
map, and denote \[H_t(x):=H(t,x),
\qquad t\in[0,1].\] If \(T\in\mathcal{R}_k(K;\mathbb{Z}_2),\) then \[\partial H_{\#}(\llbracket[0,1]\rrbracket\times T)
=
(H_1)_{\#}T-(H_0)_{\#}T
-
H_{\#}(\llbracket[0,1]\rrbracket\times \partial T).\] Since we work with \(\mathbb{Z}_2\)-coefficients, the signs are irrelevant, and we may write \[(H_1)_{\#}T-(H_0)_{\#}T
=
\partial H_{\#}(\llbracket[0,1]\rrbracket\times T)
+
H_{\#}(\llbracket[0,1]\rrbracket\times \partial T).\]
In particular, if \(H\) is a homotopy of pairs \[H:[0,1]\times(K,A)\to(K',A'),\] namely \[H(t,K)\subset K',
\qquad
H(t,A)\subset A',\] then \(H_0\) and \(H_1\) induce the same map on relative homology. Indeed, if \(T\in Z_k(K,A;\mathbb{Z}_2),\) then \[(H_1)_{\#}T-(H_0)_{\#}T
=
\partial Q+R,\] where \[Q=H_{\#}(\llbracket[0,1]\rrbracket\times T)
\in\mathcal{R}_{k+1}(K';\mathbb{Z}_2),\] and \[R=H_{\#}(\llbracket[0,1]\rrbracket\times \partial T)
\in\mathcal{R}_k(A';\mathbb{Z}_2).\] Thus \[[(H_1)_{\#}T]=[(H_0)_{\#}T]
\quad
\text{in }H_k(K',A';\mathbb{Z}_2).\]
We next recall the varifold notation used in the paper. Let \(N^{n+1}\) be a smooth Riemannian manifold. We denote by \(G_n(N)\) the Grassmannian bundle of unoriented \(n\)-planes in \(TN\). An \(n\)-varifold \(V\) in \(N\) is a Radon measure on \(G_n(N)\). Its weight measure is denoted by \(\|V\|.\) Thus, for a Borel set \(B\subset N\), \[\|V\|(B):=V(\{(x,S)\in G_n(N):x\in
B\}).\]
If \(T\) is an \(n\)-dimensional rectifiable current, we denote by \(|T|\) the associated rectifiable varifold. Its weight measure agrees with the mass
measure of \(T\), that is \(\||T|\|=\|T\|.\) In particular, \[\| |T| \|(N)=\mathbf{M}(T).\] If \(\Sigma^n\subset N\) is a
smooth embedded hypersurface, we write \(|\Sigma|\) for the varifold associated with \(\Sigma\), namely \[|\Sigma|(\varphi)
=
\int_\Sigma \varphi(x,T_x\Sigma)\,d\mathcal{H}^n(x)\] for every \(\varphi\in C_c(G_n(N))\).
For a \(C^1\) vector field \(X\) on \(N\), the first variation of an \(n\)-varifold \(V\) is defined by \[\delta V(X)
:=
\int_{G_n(N)} \operatorname{div}_S X(x)\,dV(x,S),\] where \(\operatorname{div}_S X(x)
:=
\operatorname{tr}_S(DX(x))\) is the tangential divergence of \(X\) along the \(n\)-plane \(S\). Equivalently, if \(e_1,\ldots,e_n\) is an orthonormal basis of \(S\), then \[\operatorname{div}_S X(x)
=
\sum_{i=1}^n \langle \nabla_{e_i}X,e_i\rangle.\]
We say that \(V\) has generalized mean curvature \(H\) in an open set \(U\) if \[H\in L^1_{\mathrm{loc}}(\|V\|\llcorner
U)\qquad\text{and}\qquad\delta V(X)
=
-\int_U \langle X,H\rangle\,d\|V\|\] for every compactly supported \(C^1\) vector field \(X\) in \(U\). In this paper we shall also use the
corresponding free-boundary version, where the above identity is tested only against vector fields tangent to the ambient boundary, this will be recalled in the next subsection.
We shall frequently use the \(n\)-dimensional Jacobian of a linear map on an \(n\)-plane. Let \(A:E\to F\) be a linear map between finite-dimensional
inner product spaces, and let \(S\subset E\) be an \(n\)-dimensional subspace. Choose an orthonormal basis \(e_1,\ldots,e_n\) of \(S\). We define \[J_n(A|_S)
:=
|Ae_1\wedge\cdots\wedge Ae_n|.\] Equivalently, \[J_n(A|_S)
=
\sqrt{\det(\langle Ae_i,Ae_j\rangle)_{i,j=1}^n}.\] This definition is independent of the chosen orthonormal basis.
If \(f:U\to N'\) is differentiable at \(y\in U\), and if \(S\subset T_yU\) is an \(n\)-plane, we write \[J_nf(y,S)
:=
J_n(Df(y)|_S).\] Equivalently, if \(\xi\) is a unit simple \(n\)-vector spanning \(S\), then \[J_nf(y,S)
=
|\Lambda^n Df(y)(\xi)|.\]
We shall use the standard mass estimate for push-forwards of rectifiable currents. If \(f:U\to N'\) is Lipschitz and \(T\) is an \(n\)-dimensional
rectifiable current supported in \(U\), then \[\mathbf{M}(f_\#T)
\le
\int J_nf(y,\operatorname{Tan}(T,y))\,d\|T\|(y),\] where \(\operatorname{Tan}(T,y)\) denotes the approximate tangent \(n\)-plane of \(T\) at \(y\), defined for \(\|T\|\)-almost every \(y\). In particular, \[\mathbf{M}(f_\#T)
\le
\operatorname{Lip}(f)^n\mathbf{M}(T).\] The same estimate applies to the associated rectifiable varifolds.
These are standard facts in geometric measure theory; see Federer [24] and Simon [23].
The following notions are standard in the free-boundary setting. See for example [[12]][22][14].
We now recall the class of admissible vector fields in the free-boundary setting. Let \((N^{n+1},g)\) be a smooth Riemannian manifold with boundary. We denote by \(\mathfrak
X^{\partial}(N)\) the space of smooth vector fields \(X\) on \(N\) satisfying \[X(p)\in T_p\partial N
\qquad
\text{for every }p\in\partial N.\] Equivalently, if \(\mu_{\partial N}\) denotes the unit normal to \(\partial N\) in \(N\), then \[\langle X,\mu_{\partial N}\rangle=0
\qquad
\text{on }\partial N.\]
If \(X\in\mathfrak X^{\partial}(N)\), then its local flow \(\psi_t\) preserves the ambient boundary, i.e. \(\psi_t(\partial N)\subset \partial N\) for all
sufficiently small \(t\). Indeed, the vector field \(X\) is tangent to \(\partial N\) along \(\partial N\), and hence the
restriction of its flow to \(\partial N\) is the flow of the vector field \(X|_{\partial N}\) on \(\partial N\).
This is the natural class of variations for free-boundary problems. If \(\Sigma^n\subset N\) is a properly embedded hypersurface with \(\partial\Sigma\subset\partial N,\) then the flow of
any \(X\in\mathfrak X^{\partial}(N)\) sends \(\Sigma\) to a family of hypersurfaces whose boundaries remain on \(\partial N\), i.e. \(\partial(\psi_t(\Sigma))\subset\partial N.\)
An \(n\)-varifold \(V\) in \(N\) is said to have free-boundary first variation with generalized mean curvature \(H\) if
\[\delta V(X)
=
-\int_N \langle X,H\rangle\,d\|V\|\] for every compactly supported vector field \(X\in\mathfrak X^{\partial}(N).\) In particular, \(V\) is called free-boundary stationary if \(\delta V(X)=0\) for every compactly supported \(X\in\mathfrak X^{\partial}(N).\)
For a smooth properly embedded hypersurface \(\Sigma\subset N\), the free-boundary stationarity condition is equivalent to \[H_\Sigma=0
\quad\text{in }\Sigma\qquad\text{and}\qquad\Sigma\perp\partial N
\quad\text{along }\partial\Sigma.\] Indeed, if \(\eta\) denotes the outward conormal of \(\partial\Sigma\) in \(\Sigma\), then the first variation
formula gives \[\delta\Sigma(X)
=
-\int_\Sigma \langle H_\Sigma,X\rangle\,d\mathcal{H}^n
+
\int_{\partial\Sigma}\langle X,\eta\rangle\,d\mathcal{H}^{n-1}.\] Since \(X\) is arbitrary in the interior, stationarity implies \(H_\Sigma=0\). Along \(\partial\Sigma\), the admissible vector fields are arbitrary tangent vector fields to \(\partial N\). Therefore the boundary term vanishes for all such \(X\) if
and only if \(\eta\perp T\partial N,\) or equivalently \(\eta=\pm\mu_{\partial N}.\) This is precisely the free-boundary orthogonality condition.
We now recall the notion of strict stability for free-boundary minimal hypersurfaces. The following definitions and second variation formula are standard in the study of free-boundary minimal hypersurfaces. For example, see Ambrozio–Carlotto–Sharp [14] and Li–Zhou [22].
Let \((N^{n+1},g)\) be a smooth Riemannian manifold with boundary, and let \(\Sigma^n\subset N\) be a compact, two-sided, properly embedded free-boundary minimal hypersurface. Let \(\nu\) be a global unit normal vector field along \(\Sigma\). We denote by \(A_\Sigma\) the second fundamental form of \(\Sigma\), and by \(h^{\partial N}\) the second fundamental form of \(\partial N\) in \(N\).
For a smooth function \(u\in C^\infty(\Sigma)\), the free-boundary second variation quadratic form is \[Q(u,u)
=
\int_\Sigma
\left(
|\nabla^\Sigma u|^2
-
\bigl(|A_\Sigma|^2+\operatorname{Ric}_N(\nu,\nu)\bigr)u^2
\right)
\,d\mathcal{H}^n
-
\int_{\partial\Sigma}
h^{\partial N}(\nu,\nu)u^2
\,d\mathcal{H}^{n-1}.\] Equivalently, if \[L_\Sigma u
=
\Delta_\Sigma u
+
\bigl(|A_\Sigma|^2+\operatorname{Ric}_N(\nu,\nu)\bigr)u\] denotes the Jacobi operator, then \(Q\) is the quadratic form associated with \(L_\Sigma\) together with the free-boundary
Robin boundary condition \[\partial_\eta u
+
h^{\partial N}(\nu,\nu)u
=
0
\qquad
\text{on }\partial\Sigma,\] where \(\eta\) is the outward conormal of \(\partial\Sigma\) in \(\Sigma\).
We say that \(\Sigma\) is stable if \(Q(u,u)\ge 0\) for every \(u\in C^\infty(\Sigma).\) And we say that \(\Sigma\) is
strictly stable if \(Q(u,u)>0\) for every nonzero \(u\in C^\infty(\Sigma).\) Equivalently, the first eigenvalue of the free-boundary Jacobi operator is positive.
In Section 3.5, we shall use the standard consequence that strict stability implies strict local minimality among sufficiently small free-boundary graphs over \(\Sigma\). This follows from the
Taylor expansion of the free-boundary area functional and the positivity of the second variation.
In this section, we work with a compact smooth Riemannian manifold with boundary \((N^{n+1},g)\) and \(\Sigma^n\subset N\) be a compact, two-sided, properly embedded free-boundary minimal
hypersurface, thus \(\partial \Sigma\subset \partial N\) and \(\Sigma\) meets \(\partial N\) orthogonally along \(\partial\Sigma\). Here properly embedded means that \[\Sigma\cap\partial N=\partial\Sigma.\] We also assume that \(\Sigma\) is strictly stable for area
functional.
The proof of Theorem 4 is divided into several steps. We first construct a tubular neighborhood adapted to the free-boundary condition. We then choose, in each such neighborhood, a
relative mass minimizer in the class of \(\llbracket\Sigma\rrbracket\). The main task is to show that these minimizers converge back to \(\Sigma\) with multiplicity one and, after applying
regularity theory, are actually free-boundary graphs over \(\Sigma\). The strict stability of \(\Sigma\) then rules out every nontrivial graph.
The first step is to choose a tubular neighborhood which is compatible with the ambient boundary. In the free-boundary setting, the ordinary nearest-point projection is not the most convenient object, because we need the projection, the retraction, and
the homotopies used later to preserve the boundary \(\partial N\). We therefore construct a defining function whose gradient is tangent to \(\partial N\) along \(\partial N\). Its flow gives a tubular parametrization adapted to the free-boundary condition.
Lemma 1.
There exist an open neighborhood \(\mathcal{U}\) of \(\Sigma\) in \(N\) and a smooth function \(\tau:\mathcal{U}\to
\mathbb{R}\) such that \[\Sigma=\tau^{-1}(0),
\qquad
d\tau\neq 0 \quad on\mathcal{U},\] and \[\partial_{\mu_{\partial N}}\tau=0
\quad
on\mathcal{U}\cap \partial N,\] where \(\mu_{\partial N}\) denotes the unit inward normal to \(\partial N\) in \(N\). Equivalently, \[\nabla \tau \in T\partial N
\quad
on\mathcal{U}\cap \partial N .\]
Proof. We first construct \(\tau\) locally and then patch the local functions.
Let \(p\in \partial\Sigma\). Since \(\Sigma\) is properly embedded and meets \(\partial N\) orthogonally, we may choose local coordinates \((z_1,\ldots,z_{n-1},y,u)\) near \(p\), where \[\partial N=\{u=0\},
\quad
N=\{u\ge 0\},\quad\text{and}\quad\partial\Sigma=\{y=0,\;u=0\}.\] Here \(z=(z_1,\ldots,z_{n-1})\) are coordinates along \(\partial\Sigma\), \(y\) is
the direction in \(\partial N\) normal to \(\partial\Sigma\), and \(u\) is the boundary collar coordinate. By the implicit function theorem, after shrinking
the coordinate neighborhood(denote \(U_p\)) if necessary, \(\Sigma\) can be written as \[\Sigma=\{y=f(z,u)\}\] for a smooth function \(f\). Since \(\partial\Sigma\subset \partial N\) is given by \(y=0\) when \(u=0\), we have \(f(z,0)=0.\) The free-boundary orthogonality condition implies \(\partial_u f(z,0)=0.\) Therefore the local function \(\tau_p(z,y,u):=y-f(z,u)\) satisfies \[\tau_p=0 \quad \text{on } \Sigma\qquad\text{and}\qquad\tau_p \neq 0 \text{ by } \partial_y \tau_p=1.\] Furthermore, \[\partial_u \tau_p(z,y,0)
=
-\partial_u f(z,0)
=
0.\] Thus \[\partial_{\mu_{\partial N}}\tau_p=0
\quad
on\partial N\] in this coordinate patch.
For points \(p\in \Sigma\setminus\partial\Sigma\), we choose ordinary local coordinates in the interior of \(N\) and take a local defining function \(\tau_p\) for \(\Sigma\). Since \(\Sigma\) is two-sided, the local defining functions can be chosen with consistent sign, i.e. \(d\tau_p(\nu_\Sigma) > 0\) on \(U_p\cap \Sigma\), where \(\nu_\Sigma\) is the global unit normal on \(\Sigma\).
Take a finite open cover \(\{U_\alpha\}\) of \(\Sigma\) by such coordinate neighborhoods. Choose a smooth partition of unity \(\{\chi_\alpha\}\)
subordinate to \(\{U_\alpha\}\), with the following additional property near \(\partial N\): \[\partial_{\mu_{\partial N}}\chi_\alpha=0
\quad
on\partial N.\] This is obtained by first taking a partition of unity on \(\partial N\), extending it constantly along the boundary collar direction, and then adding interior cutoff functions supported away from
\(\partial N\). Define \(\tau:=\sum_\alpha \chi_\alpha \tau_\alpha.\) Since each \(\tau_\alpha\) vanishes on \(\Sigma\), we
have \[\tau=0
\quad
on\Sigma.\] Moreover, along \(\Sigma\), \[d\tau
=
\sum_\alpha \chi_\alpha d\tau_\alpha,\] because the terms \(d\chi_\alpha\,\tau_\alpha\) vanish on \(\Sigma\). By \(d\tau_\alpha(\nu_\Sigma) > 0\)
for each \(\alpha\), this is nonzero on \(\Sigma\). Shrinking \(\mathcal{U}\) if necessary, we may assume \[d\tau\neq 0
\quad
on\mathcal{U}.\]
Finally, on \(\mathcal{U}\cap\partial N\), each local defining function satisfies \(\partial_{\mu_{\partial N}}\tau_\alpha=0,\) and the partition functions satisfy \(\partial_{\mu_{\partial N}}\chi_\alpha=0.\) Therefore \[\partial_{\mu_{\partial N}}\tau
=
\sum_\alpha
(\partial_{\mu_{\partial N}}\chi_\alpha)\tau_\alpha
+
\sum_\alpha
\chi_\alpha(\partial_{\mu_{\partial N}}\tau_\alpha)
=
0
\quad
on\mathcal{U}\cap\partial N.\] This proves the lemma. ◻
Remark 6. The condition \(\partial_{\mu_{\partial N}}\tau=0\) is precisely what makes the following construction compatible with the free boundary. Indeed, it implies that \(\nabla\tau\), and hence the vector field \(Z\) defined below, is tangent to \(\partial N\) along the ambient boundary.
We now define a transverse vector field by \[Z:=\frac{\nabla \tau}{|\nabla \tau|^2}.\] Here \(\nabla\tau\) is the gradient with respect to the ambient metric \(g\). Since \(d\tau\neq0\) on \(U\), this vector field is smooth. Moreover, \[d\tau(Z)
=
g\left(\nabla\tau,\frac{\nabla\tau}{|\nabla\tau|^2}\right)
=
1.\] Because \(\nabla\tau\in T\partial N\) on \(U\cap\partial N\), we also have \[Z\in T\partial N
\quad
onU\cap\partial N.\] Thus the flow of \(Z\) preserves \(\partial N\). Let \(\Phi_s\) denote the associated local flow of \(Z\). Define \[F:\Sigma\times (-r_0,r_0)\to N \qquad \text{by }F(x,s):=\Phi_s(x).\] After shrinking \(r_0>0\) if necessary, the map \(F\) is a diffeomorphism onto its image. Indeed, \(F(x,0)=x,\) and \(\partial_s F(x,0)=Z(x).\) Since \(\tau=0\) on \(\Sigma\) and \(d\tau(Z)=1\), the vector \(Z(x)\) is transverse to \(T_x\Sigma\). Hence \[D
F_{(x,0)}:T_x\Sigma\oplus \mathbb{R}\to T_xN\] is an isomorphism. The inverse function theorem gives the local result, and compactness of \(\Sigma\) gives a uniform \(r_0\).
Furthermore, since \(Z\) is tangent to \(\partial N\), its flow preserves \(\partial N\). Hence \[F(\partial\Sigma\times(-r_0,r_0))\subset \partial N.\] For \(x\in\Sigma\), we compute \[\frac{d}{ds}\tau(F(x,s))
=
d\tau(\partial_sF(x,s))
=
d\tau(Z(F(x,s)))
=
1.\] Since \(\tau(F(x,0))=0\), it follows that \(\tau(F(x,s))=s\) is exactly the height function of \(F\). Therefore \[F(\Sigma\times\{s\})=\{\tau=s\}\cap F(\Sigma\times(-r_0,r_0)).\] We denote \[\Sigma_s:=F(\Sigma\times\{s\})=\{\tau=s\}\cap \mathcal{U}.\] Since \(T\Sigma_s=\ker
d\tau,\) and \(Z=\frac{\nabla\tau}{|\nabla\tau|^2},\) we have \(Z\perp T\Sigma_s.\) Consequently the foliation \(\{\Sigma_s\}_{|s|<r_0}\) is
transverse and orthogonal to the flow direction \(Z\). For \(0<r<r_0\), define \[K_r:=F(\Sigma\times[-r,r])\qquad A_r:=K_r\cap \partial N.\] Since the
flow preserves \(\partial N\), we have \(A_r=F(\partial\Sigma\times[-r,r]).\) We also define the tubular projection \[\pi_r:K_r\to \Sigma \qquad\text{by
}\pi_r(F(x,s))=x.\]
Finally, for later use, define the retraction \[P_r:F(\Sigma\times(-r_0,r_0))\to K_r\qquad\text{by }P_r(F(x,s)):=F(x,q_r(s)),\] where \[q_r(s):=
\begin{cases}
-r, & s<-r,\\
s, & -r\le s\le r,\\
r, & s>r.
\end{cases}\] Then \[P_r|_{K_r}=\operatorname{id}_{K_r},\qquad P_r(\partial N\cap \mathcal{U})\subset \partial N.\] Although \(P_r\) is generally not smooth along the hypersurfaces
\(\{s=\pm r\}\), it is Lipschitz. Hence it induces push-forwards on currents and varifolds.
3.2 Convergence as \(\mathbb{Z}_2\)-Currents and Varifolds↩︎
We now pass from the geometric neighborhood to the variational problem inside it. For each small \(r\), we minimize mass in the relative homology class of \(\llbracket\Sigma\rrbracket\)
inside the pair \((K_r,A_r)\). The goal of this subsection is to prove that these minimizers converge back to \(\llbracket\Sigma\rrbracket\) as relative currents and to \(|\Sigma|\) as varifolds.
In this subsection, We consider the class \[\mathcal{C}_r
:=
\left\{
T\in \mathcal{Z}_n(K_r,A_r;\mathbb{Z}_2):
[T]=[\Sigma]\;\text{in } H_n(K_r,A_r;\mathbb{Z}_2)
\right\}.\] Define \[m_r:=\inf\{\mathbf{M}(T):T\in\mathcal{C}_r\}.\] By compactness of integral currents modulo \(2\), there exists \(T_r\in\mathcal{C}_r\) such that \(\mathbf{M}(T_r)=m_r.\) Equivalently, \[\mathbf{M}(T_r)
=
\inf
\left\{
\mathbf{M}(T):
T\in \mathcal{Z}_n(K_r,A_r;\mathbb{Z}_2),
\;[T]=[\Sigma]\in H_n(K_r,A_r;\mathbb{Z}_2)
\right\}.\] We call \(T_r\) a mass-minimizing relative mod \(2\) representative of \([\Sigma]\) in \((K_r,A_r)\), and
will show that:
\(T_r\rightarrow \llbracket \Sigma \rrbracket\) as mod 2 currents, where \(\llbracket \Sigma \rrbracket\) is the mod 2 current induced by \(\Sigma\).
\(V_r:=|T_r|\rightarrow |\Sigma|\) as varifolds, where \(|\Sigma|\) is the varifold induced by \(\Sigma\). Moreover, the varifold convergence has
multiplicity 1.
The key point is that the relative homology constraint is strong enough to control the projection of \(T_r\) onto \(\Sigma\). This will give the lower mass bound needed to force
convergence.
Before we start, we need to prove some useful results. First we show that \((\pi_r)_{\#} T_r=\llbracket \Sigma \rrbracket.\)
Lemma 2. With \(\Sigma,\;A_r,\;K_r,\;\pi_r\) defined as in Section 1, \(\llbracket \Sigma \rrbracket\) represents the mod 2 current induced by \(\Sigma\), note that \(\pi_r\) satisfies: \[\pi_r|_{\Sigma}=\operatorname{id}_{\Sigma},
\qquad
\pi_r(A_r)\subset \partial\Sigma.\] Let \(T_r\in \mathcal{Z}_n(K_r,A_r;\mathbb{Z}_2)\) satisfy \[[T_r]=[\llbracket\Sigma\rrbracket]
\quad\text{in}\quad
H_n(K_r,A_r;\mathbb{Z}_2).\] Then we have \((\pi_r)_{\#}T_r=\llbracket\Sigma\rrbracket\) as \(\mathbb{Z}_2\)-currents.
Proof. Since \([T_r]=[\llbracket\Sigma\rrbracket]\) in \(H_n(K_r,A_r;\mathbb{Z}_2),\) there exist \[Q_r\in \mathcal{I}_{n+1}(K_r;\mathbb{Z}_2),
\qquad
R_r\in \mathcal{I}_n(A_r;\mathbb{Z}_2)\] such that \[T_r-\llbracket\Sigma\rrbracket
=
\partial Q_r+R_r.\] Here and below the sign is irrelevant over \(\mathbb{Z}_2\).
Applying \((\pi_r)_{\#}\) to both sides gives \[(\pi_r)_{\#}T_r-(\pi_r)_{\#}\llbracket\Sigma\rrbracket
=
\partial(\pi_r)_{\#}Q_r+(\pi_r)_{\#}R_r.\] Since \(\pi_r|_{\Sigma}=\operatorname{id}_{\Sigma},\) we have \((\pi_r)_{\#}\llbracket\Sigma\rrbracket
=
\llbracket\Sigma\rrbracket.\) Moreover, since \(\operatorname{spt}R_r\subset A_r\) and \(\pi_r(A_r)\subset \partial\Sigma,\) we have \(\operatorname{spt}(\pi_r)_{\#}R_r\subset\partial\Sigma.\) Thus \[(\pi_r)_{\#}T_r-\llbracket\Sigma\rrbracket
=
\partial(\pi_r)_{\#}Q_r+(\pi_r)_{\#}R_r,\] with the last term supported in \(\partial\Sigma\). Hence \[[(\pi_r)_{\#}T_r]
=
[\llbracket\Sigma\rrbracket]
\quad\text{in}\quad
H_n(\Sigma,\partial\Sigma;\mathbb{Z}_2).\]
Now \((\pi_r)_{\#}T_r\) is an \(n\)-dimensional relative cycle in \((\Sigma,\partial\Sigma)\). Since it is supported in the \(n\)-manifold \(\Sigma\), the constancy theorem implies that \[(\pi_r)_{\#}T_r
=
\sum_{\alpha} c_{\alpha}\llbracket\Sigma_{\alpha}\rrbracket,
\qquad
c_{\alpha}\in\mathbb{Z}_2,\] where \(\{\Sigma_{\alpha}\}\) are possibly connected components of \(\Sigma\).
The equality \[[(\pi_r)_{\#}T_r]
=
[\llbracket\Sigma\rrbracket]
\quad\text{in}\quad
H_n(\Sigma,\partial\Sigma;\mathbb{Z}_2)\] forces \(c_{\alpha}=1\) for every component \(\Sigma_{\alpha}\). Therefore \[(\pi_r)_{\#}T_r
=
\sum_{\alpha}\llbracket\Sigma_{\alpha}\rrbracket
=
\llbracket\Sigma\rrbracket.\] ◻
Thus the projection does not lose the relative fundamental class. This is the homological input which replaces any pointwise graphical information about \(T_r\). Although \(T_r\) may be
singular or non-graphical, its projection already has exactly the same mod \(2\) multiplicity as \(\Sigma\).
To convert the projected homology information into a mass estimate, we need to control how much the tubular projection can increase \(n\)-dimensional area. Since \(K_r\) collapses to
\(\Sigma\) as \(r\to0\), the differential of \(\pi_r\) should be close to the orthogonal projection onto \(T\Sigma\). The
following lemma makes this quantitative.
Lemma 3 (Jacobian estimate for the tubular projection). Let \[F:\Sigma\times(-r_0,r_0)\to N,
\qquad
F(x,t)=\Phi_t(x),\] be an admissible free-boundary tubular parametrization of \(\Sigma\) as in Section 1, and
let \[\pi_r:F(\Sigma\times[-r,r])\to\Sigma\] be the associated projection \(\pi_r(F(x,t))=x.\) Then, after decreasing \(r_0>0\) if necessary, there
exists a constant \(C>0\), independent of \(r\), such that for every \(0<r<r_0\), every \[y\in
K_r:=F(\Sigma\times[-r,r]),\] and every \(n\)-plane \[S\subset T_yN,\] one has \[J_n\pi_r(y,S)\le 1+Cr.\] Here \(J_n\pi_r(y,S)\) denotes the \(n\)-dimensional Jacobian of \(D\pi_r(y)\) restricted to \(S\).
Proof. Write \[y=F(x,t),
\qquad
x\in\Sigma,
\qquad
|t|\le r.\] We first prove that \(D\pi_r(y)\) is uniformly close to the orthogonal projection onto \(T_x\Sigma.\)
Let \[\mathcal{P}_{x,t}:T_{F(x,t)}N\to T_xN\] denote parallel transport along the curve \[s\mapsto F(x,st),
\qquad 0\le s\le1.\] Define \[A_{x,t}:=D\pi_r(F(x,t))\circ \mathcal{P}_{x,t}^{-1}:T_xN\to T_x\Sigma.\] Since \(F\) and \(\pi_r\) are smooth and \(\Sigma\) is compact, the map \((x,t)\mapsto A_{x,t}\) is smooth for \(|t|<r_0\). At \(t=0\), we have \(F(x,0)=x\), and by definition \(\pi_r(F(x,t))=x.\) Hence \[D\pi_r(x)|_{T_x\Sigma}=\operatorname{id}_{T_x\Sigma},
\qquad
D\pi_r(x)(\nu_\Sigma(x))=0.\] Therefore \(A_{x,0}=\operatorname{pr}_{T_x\Sigma},\) where \(\operatorname{pr}_{T_x\Sigma}:T_xN\to T_x\Sigma\) is the orthogonal projection.
By smoothness and compactness, there is a constant \(C_1>0\) such that \[\|A_{x,t}-\operatorname{pr}_{T_x\Sigma}\|\le C_1|t|\] for all \(x\in\Sigma\)
and all \(|t|<r_0\). In particular, since \(|t|\le r\), \[\|A_{x,t}-\operatorname{pr}_{T_x\Sigma}\|\le C_1r.\]
Let \(S\subset T_yN\) be an \(n\)-plane and set \[\widetilde{S}:=\mathcal{P}_{x,t}(S)\subset T_xN.\] Since parallel transport is an isometry, \[J_n\pi_r(y,S)
=
J_n(A_{x,t},\widetilde{S}).\] Let \(\xi\) be a unit simple \(n\)-vector spanning \(\widetilde{S}\). Then \[J_n(A_{x,t},\widetilde{S})
=
|\Lambda^n A_{x,t}(\xi)|.\] The map \(A\mapsto \Lambda^n A\) is locally Lipschitz on bounded subsets of \(\operatorname{Hom}(T_xN,T_x\Sigma)\). Since \(A_{x,t}\) and \(\operatorname{pr}_{T_x\Sigma}\) are uniformly bounded, there exists \(C_2>0\) such that \[|\Lambda^n A_{x,t}(\xi)|
\le
|\Lambda^n \operatorname{pr}_{T_x\Sigma}(\xi)|
+
C_2\|A_{x,t}-\operatorname{pr}_{T_x\Sigma}\|.\] Thus \[J_n\pi_r(y,S)
\le
J_n(\operatorname{pr}_{T_x\Sigma},\widetilde{S})+C_2C_1r.\] Since \(\operatorname{pr}_{T_x\Sigma}\) is an orthogonal projection, it is \(1\)-Lipschitz, and hence \(J_n(\operatorname{pr}_{T_x\Sigma},\widetilde{S})\le1.\) Therefore we have \(J_n\pi_r(y,S)\le 1+Cr\) for some constant \(C>0\), independent of \(r\). This proves the lemma. ◻
With Lemmas 2 and 3, it suffices to show the mass convergence.
Lemma 4 (Mass convergence). Let \(T_r\) be a relative mass minimizer in \(K_r\) in the class of \(\llbracket\Sigma\rrbracket\), namely
\[[T_r]=[\llbracket\Sigma\rrbracket]
\quad\text{in}\quad
H_n(K_r,A_r;\mathbb{Z}_2),\] where \(K_r,\;
A_r\) as defined in Section 1. Assume that \((\pi_r)_{\#}T_r=\llbracket\Sigma\rrbracket.\) Then \[\mathbf{M}(T_r)\to \mathbf{M}(\llbracket\Sigma\rrbracket)
=
\mathcal{H}^n(\Sigma)\] as \(r\to0\). More precisely, there exists \(C>0\) independent of \(r\) such that \[\frac{1}{1+Cr}\mathbf{M}(\llbracket\Sigma\rrbracket)
\le
\mathbf{M}(T_r)
\le
\mathbf{M}(\llbracket\Sigma\rrbracket).\]
Proof. Since \(T_r\) minimizes mass in its relative homology class and since \(\llbracket\Sigma\rrbracket\) itself is an admissible competitor, we immediately have \(\mathbf{M}(T_r)\le \mathbf{M}(\llbracket\Sigma\rrbracket).\)
We now prove the reverse inequality up to a ratio \(1+Cr\). Since \((\pi_r)_{\#}T_r=\llbracket\Sigma\rrbracket,\) we have \[\mathbf{M}(\llbracket\Sigma\rrbracket)
=
\mathbf{M}((\pi_r)_{\#}T_r).\] By the mass estimate for push-forwards of rectifiable currents, \[\mathbf{M}((\pi_r)_{\#}T_r)
\le
\int J_n\pi_r\bigl(y,\operatorname{Tan}(T_r,y)\bigr)\,d\|T_r\|(y),\] where \(\operatorname{Tan}(T_r,y)\) denotes the approximate tangent \(n\)-plane of \(T_r\) at \(y\), defined for \(\|T_r\|\)-almost every \(y\).
By the Jacobian estimate from Lemma 3, \[J_n\pi_r\bigl(y,\operatorname{Tan}(T_r,y)\bigr)\le 1+Cr.\] Therefore \[\mathbf{M}(\llbracket\Sigma\rrbracket)
\le
(1+Cr)\int d\|T_r\|
=
(1+Cr)\mathbf{M}(T_r).\] Hence \[\mathbf{M}(T_r)
\ge
\frac{1}{1+Cr}\mathbf{M}(\llbracket\Sigma\rrbracket).\]
Combining this with \(\mathbf{M}(T_r)\le \mathbf{M}(\llbracket\Sigma\rrbracket)\) gives \[\frac{1}{1+Cr}\mathbf{M}(\llbracket\Sigma\rrbracket)
\le
\mathbf{M}(T_r)
\le
\mathbf{M}(\llbracket\Sigma\rrbracket).\] Letting \(r\to0\), we obtain \(\mathbf{M}(T_r)\to \mathbf{M}(\llbracket\Sigma\rrbracket).\) ◻
Hence by above lemma, any relative minimizer has almost the same mass as \(\Sigma\) when the tubular neighborhood is sufficiently thin.
We first record the convergence in the relative flat topology. The proof uses the deformation retraction from \(K_r\) to \(\Sigma\) along the flow lines of \(Z\). Since this homotopy preserves \(A_r\), the homotopy formula gives a relative filling of \(T_r-\llbracket\Sigma\rrbracket\). The filling has small mass
because the homotopy moves points by distance \(O(r)\).
Lemma 5 (Relative flat convergence). With above \(K_r,\;A_r,\;F,\;\pi_r\), Assume that \((\pi_r)_{\#}T_r=\llbracket \Sigma\rrbracket.\) Then we have: \[T_r\to \llbracket\Sigma\rrbracket\] in the relative flat topology of the pair \((K_r,A_r)\). More precisely, there exists a constant \(C>0\), independent of
\(r\), such that \[\mathcal{F}_{\mathrm{rel}}(T_r-\llbracket\Sigma\rrbracket)
\le
Cr\,\mathbf{M}(T_r).\] In particular, since \(\mathbf{M}(T_r)\le \mathbf{M}(\llbracket\Sigma\rrbracket)\), one has \[\mathcal{F}_{\mathrm{rel}}(T_r-\llbracket\Sigma\rrbracket)\to0
\qquad\text{as }r\to0.\]
Proof. Define a deformation retraction \(H:[0,1]\times K_r\to K_r\) by \[H(s,F(x,t))=F(x,(1-s)t).\] Then \[H(0,y)=y,
\qquad
H(1,y)=\pi_r(y).\] Moreover, since the tubular neighborhood is admissible with respect to the free boundary, \[H(s,A_r)\subset A_r\qquad\text{for every \(s\in[0,1]\).}\]
Let \(S_r=H_{\#}(\llbracket[0,1]\rrbracket\times T_r).\) By the homotopy formula for currents, \[\partial S_r
=
(H_1)_{\#}T_r-(H_0)_{\#}T_r
-
H_{\#}(\llbracket[0,1]\rrbracket\times \partial T_r).\] Here \[H_0=\operatorname{id}_{K_r},
\qquad
H_1=\pi_r.\] Thus \[\partial S_r
=
(\pi_r)_{\#}T_r-T_r
-
H_{\#}(\llbracket[0,1]\rrbracket\times \partial T_r).\] Since \((\pi_r)_{\#}T_r=\llbracket\Sigma\rrbracket,\) we obtain \[T_r-\llbracket\Sigma\rrbracket
=
-\partial S_r
-
H_{\#}(\llbracket[0,1]\rrbracket\times \partial T_r).\] Over \(\mathbb{Z}_2\), the signs are irrelevant. Hence \[T_r-\llbracket\Sigma\rrbracket
=
\partial S_r+R_r,\] where \(R_r=H_{\#}(\llbracket[0,1]\rrbracket\times \partial T_r).\)
Since \(T_r\) is a relative cycle in \((K_r,A_r)\), we have \(\operatorname{spt}\partial T_r\subset A_r.\) And since \(H(s,A_r)\subset A_r\), it follows that \(\operatorname{spt}R_r\subset A_r.\) Therefore \[T_r-\llbracket\Sigma\rrbracket
=
\partial S_r+R_r,
\qquad
\operatorname{spt}R_r\subset A_r.\] This shows that \(T_r-\llbracket\Sigma\rrbracket\) is a relative boundary up to the filling \(S_r\).
It remains to estimate \(\mathbf{M}(S_r)\). We claim that \[\mathbf{M}(S_r)\le Cr\,\mathbf{M}(T_r).\] Indeed, write \[y=F(x,t),
\qquad |t|\le r.\] Then \(H(s,y)=F(x,(1-s)t).\) Differentiating in \(s\), we get \[\partial_s H(s,F(x,t))
=
-t\,\partial_tF(x,(1-s)t).\] Since \(F\) is a smooth tubular parametrization and \(r\) is sufficiently small, there is a constant \(C_1>0\),
independent of \(r\), such that \(|\partial_tF|\le C_1.\) Therefore \[|\partial_s H(s,F(x,t))|\le C_1|t|\le C_1r.\] Moreover, the derivatives of \(H\) in the spatial directions are uniformly bounded: \[|D_yH(s,y)|\le C_2.\] Hence the \((n+1)\)-Jacobian of \(H\) on the
product current \(\llbracket[0,1]\rrbracket\times T_r\) is bounded by \(J_{n+1}H\le Cr.\) Consequently, by the mass estimate for push-forwards, \[\mathbf{M}(S_r)
=
\mathbf{M}\bigl(H_{\#}(\llbracket[0,1]\rrbracket\times T_r)\bigr)
\le
\int_{\llbracket[0,1]\rrbracket\times T_r} J_{n+1}H
\le
Cr\,\mathbf{M}(T_r).\]
By the definition of the relative flat norm, since \[T_r-\llbracket\Sigma\rrbracket=\partial S_r+R_r,
\qquad
\operatorname{spt}R_r\subset A_r,\] we have \[\mathcal{F}_{\mathrm{rel}}(T_r-\llbracket\Sigma\rrbracket)
\le
\mathbf{M}(S_r).\] Therefore \[\mathcal{F}_{\mathrm{rel}}(T_r-\llbracket\Sigma\rrbracket)
\le
Cr\,\mathbf{M}(T_r).\] Finally, since \(T_r\) is mass minimizing in its relative class and \(\llbracket\Sigma\rrbracket\) is an admissible competitor, one has \(\mathbf{M}(T_r)\le \mathbf{M}(\llbracket\Sigma\rrbracket).\) Thus \[\mathcal{F}_{\mathrm{rel}}(T_r-\llbracket\Sigma\rrbracket)
\le
Cr\,\mathbf{M}(\llbracket\Sigma\rrbracket)
\to0 \qquad\text{as }r\to0.\] ◻
We next upgrade the mass and relative flat convergence to varifold convergence. The support of \(T_r\) lies in \(K_r\), and hence any varifold limit is supported on \(\Sigma\). The mass convergence shows that the limiting mass is exactly \(\mathcal{H}^n(\Sigma)\). It remains only to show that no tangent-plane oscillation is lost in the limit.
The basic Jacobian estimate \(J_n\pi_r\le 1+Cr\) gives mass convergence. A slightly sharper version also detects the tilt of \(T_r\) away from \(T\Sigma\): if a tangent plane is not close to \(T\Sigma\), the orthogonal projection strictly decreases its \(n\)-dimensional volume. This gives the tilt-excess
convergence.
Lemma 6 (Tilt-excess convergence). With \(K_r,\;A_r,\;\pi_r:K_r\to \Sigma\) same definition as above. Let \(T_r\in \mathcal{Z}_n(K_r,A_r;\mathbb{Z}_2),\) be a relative
mass minimizer in the relative homology class of \(\llbracket\Sigma\rrbracket\), and set \(V_r=|T_r|.\) Assume that \((\pi_r)_{\#}T_r=\llbracket\Sigma\rrbracket.\) Then \[\int
\operatorname{dist}_{\mathcal{G}}
\left(
\operatorname{Tan}(T_r,y),
T_{\pi_r(y)}\Sigma
\right)^2
\,d\|T_r\|(y)
\to0\] as \(r\to0\). Quantitatively, there exists a constant \(C>0\), independent of \(r\), such that \[\int
\operatorname{dist}_{\mathcal{G}}
\left(
\operatorname{Tan}(T_r,y),
T_{\pi_r(y)}\Sigma
\right)^2
\,d\|T_r\|(y)
\le Cr.\]
Proof. We first recall the definition of the Grassmannian distance \(\operatorname{dist}_{\mathcal{G}}\) appearing above. If \[y=F(x,t),
\qquad x=\pi_r(y),
\qquad |t|\le r,\] then \(\operatorname{Tan}(T_r,y)\subset T_yN\), whereas \(T_x\Sigma\subset T_xN.\) To compare these two \(n\)-planes, same as the
proof of lemma 3, let \[\mathcal{P}_{y\to x}:T_yN\to T_xN\] be parallel transport along the curve \[s\mapsto
F(x,(1-s)t),
\qquad 0\le s\le1.\] Set \[\widetilde{S}_y=\mathcal{P}_{y\to x}\bigl(\operatorname{Tan}(T_r,y)\bigr)
\subset T_xN.\] Then we define \[\operatorname{dist}_{\mathcal{G}}
\left(
\operatorname{Tan}(T_r,y),
T_x\Sigma
\right)
:=
\left\|
\operatorname{pr}_{\widetilde{S}_y}
-
\operatorname{pr}_{T_x\Sigma}
\right\|_{\operatorname{HS}},\] where \(\operatorname{pr}_{\widetilde{S}_y}\) and \(\operatorname{pr}_{T_x\Sigma}\) are the orthogonal projections onto \(\widetilde{S}_y\) and \(T_x\Sigma\) inside \(T_xN\), and \(\|\cdot\|_{\operatorname{HS}}\) denotes the Hilbert-Schmidt norm.
We now prove an improved Jacobian estimate for \(\pi_r\). We claim that there are constants \[C>0,
\qquad
c>0,\] independent of \(r\), such that for all \(y\in K_r\) and all \(n\)-planes \(S\subset T_yN\), one has \[J_n\pi_r(y,S)
\le
1+Cr
-
c\,\operatorname{dist}_{\mathcal{G}}
\left(
S,T_{\pi_r(y)}\Sigma
\right)^2.\]
Recall that we’ve shown in the proof of lemma 3 that \[J_n\pi_r(y,S)
\le
J_n(\operatorname{pr}_{T_x\Sigma},\widetilde{S})
+
Cr.\] We now use the elementary linear algebra estimate \[J_n(\operatorname{pr}_{T_x\Sigma},\widetilde{S})
\le
1
-
c_0
\operatorname{dist}_{\mathcal{G}}
\left(
\widetilde{S},T_x\Sigma
\right)^2.\] Let us briefly justify it. Since both \(\widetilde{S}\) and \(T_x\Sigma\) are \(n\)-planes in the \((n+1)\)-dimensional inner product space \(T_xN\), there is only one non-zero principal angle between them. Let this angle be \(\theta\in[0,\pi/2]\). Then \[J_n(\operatorname{pr}_{T_x\Sigma},\widetilde{S})
=
\cos\theta.\] Moreover, \[\operatorname{dist}_{\mathcal{G}}
\left(
\widetilde{S},T_x\Sigma
\right)^2
\asymp
\sin^2\theta.\] For example, if the Hilbert-Schmidt distance of orthogonal projections is used, then \[\operatorname{dist}_{\mathcal{G}}
\left(
\widetilde{S},T_x\Sigma
\right)^2
=
2\sin^2\theta.\] Since \[1-\cos\theta
=
\frac{\sin^2\theta}{1+\cos\theta}
\ge
\frac{1}{2}\sin^2\theta,\] we get \[\cos\theta
\le
1-c_0
\operatorname{dist}_{\mathcal{G}}
\left(
\widetilde{S},T_x\Sigma
\right)^2\] for some constant \(c_0>0\). This proves the linear algebra estimate.
Combining the previous estimates gives \[J_n\pi_r(y,S)
\le
1+Cr
-
c_0
\operatorname{dist}_{\mathcal{G}}
\left(
\widetilde{S},T_x\Sigma
\right)^2.\] By the definition of the Grassmannian distance between \(S\subset T_yN\) and \(T_x\Sigma\subset T_xN\), this is exactly \[J_n\pi_r(y,S)
\le
1+Cr
-
c
\operatorname{dist}_{\mathcal{G}}
\left(
S,T_{\pi_r(y)}\Sigma
\right)^2.\] This proves the Jacobian estimate.
We now apply this estimate to \(S=\operatorname{Tan}(T_r,y)\), which exists for \(\|T_r\|\)-almost every \(y\), since \(T_r\) is an integral current.
By the push-forward mass estimate for rectifiable currents, \[\mathbf{M}((\pi_r)_{\#}T_r)
\le
\int
J_n\pi_r
\left(
y,\operatorname{Tan}(T_r,y)
\right)
\,d\|T_r\|(y).\] Using \((\pi_r)_{\#}T_r=\llbracket\Sigma\rrbracket,\) we have \(\mathbf{M}(\llbracket\Sigma\rrbracket)
=
\mathbf{M}((\pi_r)_{\#}T_r).\) Therefore \[\mathbf{M}(\llbracket\Sigma\rrbracket)
\le
\int
J_n\pi_r
\left(
y,\operatorname{Tan}(T_r,y)
\right)
\,d\|T_r\|(y).\] By the Jacobian estimate, \[\mathbf{M}(\llbracket\Sigma\rrbracket)
\le
\int
\left[
1+Cr
-
c\,
\operatorname{dist}_{\mathcal{G}}
\left(
\operatorname{Tan}(T_r,y),
T_{\pi_r(y)}\Sigma
\right)^2
\right]
d\|T_r\|(y).\] Thus \[\mathbf{M}(\llbracket\Sigma\rrbracket)
\le
(1+Cr)\mathbf{M}(T_r)
-
c
\int
\operatorname{dist}_{\mathcal{G}}
\left(
\operatorname{Tan}(T_r,y),
T_{\pi_r(y)}\Sigma
\right)^2
d\|T_r\|(y).\] Since \(T_r\) is mass minimizing in its relative homology class and since \(\llbracket\Sigma\rrbracket\) is an admissible competitor, we have \(\mathbf{M}(T_r)\le \mathbf{M}(\llbracket\Sigma\rrbracket).\) Hence \[\mathbf{M}(\llbracket\Sigma\rrbracket)
\le
(1+Cr)\mathbf{M}(\llbracket\Sigma\rrbracket)
-
c
\int
\operatorname{dist}_{\mathcal{G}}
\left(
\operatorname{Tan}(T_r,y),
T_{\pi_r(y)}\Sigma
\right)^2
d\|T_r\|(y).\] Rearranging, we obtain \[c
\int
\operatorname{dist}_{\mathcal{G}}
\left(
\operatorname{Tan}(T_r,y),
T_{\pi_r(y)}\Sigma
\right)^2
d\|T_r\|(y)
\le
Cr\,\mathbf{M}(\llbracket\Sigma\rrbracket).\] Therefore \[\int
\operatorname{dist}_{\mathcal{G}}
\left(
\operatorname{Tan}(T_r,y),
T_{\pi_r(y)}\Sigma
\right)^2
d\|T_r\|(y)
\le
Cr.\] Letting \(r\to0\), we get \[\int
\operatorname{dist}_{\mathcal{G}}
\left(
\operatorname{Tan}(T_r,y),
T_{\pi_r(y)}\Sigma
\right)^2
d\|T_r\|(y)
\to0.\] This proves the tilt-excess convergence. ◻
With lemma 6 above, it suffices to prove the varifold convergence of \(V_r.\)
Corollary 1 (Varifold convergence). Under the assumptions above, \(V_r:=|T_r|\to |\Sigma|\) as varifolds.
Proof. Let \(\varphi\in C_c(G_n(N))\). We need to show \[\int \varphi(y,S)\,dV_r(y,S)
\to
\int_{\Sigma}\varphi(x,T_x\Sigma)\,d\mathcal{H}^n(x).\]
By support convergence, for \(\|T_r\|\)-almost every \(y\in\operatorname{spt}T_r\), \[d_N(y,\pi_r(y))\le Cr.\] By tilt-excess convergence, \[\int
\operatorname{dist}_{\mathcal{G}}
\left(
\operatorname{Tan}(T_r,y),
T_{\pi_r(y)}\Sigma
\right)^2
\,d\|T_r\|(y)
\to0.\] Since \(\varphi\) is uniformly continuous on the compact region, it follows that \[\int
\left|
\varphi\left(y,\operatorname{Tan}(T_r,y)\right)
-
\varphi\left(\pi_r(y),T_{\pi_r(y)}\Sigma\right)
\right|
\,d\|T_r\|(y)
\to0.\]
Thus it suffices to prove \[\int
\varphi\left(\pi_r(y),T_{\pi_r(y)}\Sigma\right)
\,d\|T_r\|(y)
\to
\int_{\Sigma}\varphi(x,T_x\Sigma)\,d\mathcal{H}^n(x).\] This follows from \((\pi_r)_{\#}T_r=\llbracket\Sigma\rrbracket\) together with the mass convergence \(\mathbf{M}(T_r)\to
\mathbf{M}(\llbracket\Sigma\rrbracket)\) and the Jacobian estimate \(J_n\pi_r\le 1+Cr.\) Indeed, the projection loses no mass in the limit, so the push-forward of the measures \(\|T_r\|\) under \(\pi_r\) converges weakly to \(\mathcal{H}^n\llcorner\Sigma.\) Therefore \[\int \varphi(y,S)\,dV_r(y,S)
\to
\int_{\Sigma}\varphi(x,T_x\Sigma)\,d\mathcal{H}^n(x).\] Hence \(V_r\to|\Sigma|\) as varifolds. Since the limiting varifold is precisely \(|\Sigma|\), rather than \(m|\Sigma|\) for some \(m\ge2\), the convergence is of multiplicity one. ◻
We have therefore shown that the relative minimizers converge to \(\Sigma\) with multiplicity one as varifolds. This is still not enough to apply regularity theory: we also need a uniform first variation bound. This is
the purpose of the next subsection.
The next goal is to prove a first variation bound for \(V_r=|T_r|\) which is uniform in \(r\). This is not immediate from the minimizing property, because \(T_r\) minimizes only among competitors supported in \(K_r\). A general free-boundary variation may move points outside \(K_r\). The idea is to compose such a
variation with the Lipschitz retraction \(P_r\) constructed in Section 3.1. This produces admissible competitors in the same relative homology class.
Lemma 7. Let \(K_r\), \(A_r,\) defined as above, and let \(T_r\in \mathcal{Z}_n(K_r,A_r;\mathbb{Z}_2)\) be a relative cycle. Let \(X\in \mathfrak{X}^\partial(N)\), namely \[X(p)\in T_p\partial N\qquad p\in\partial N,\] and let \(\psi_t\) be the flow generated by \(X\). Let \[P_r:F(\Sigma\times(-r_0,r_0))\to K_r,\qquad P_r(F(x,s))=F(x,q_r(s)),\] as in Section 3.1. For \(|t|\) sufficiently small,
set \(f_t:=P_r\circ\psi_t.\) Then \[(f_t)_\#T_r\in \mathcal{Z}_n(K_r,A_r;\mathbb{Z}_2).\] Moreover, \[[(f_t)_\#T_r]=[T_r]
\qquad
\text{in }H_n(K_r,A_r;\mathbb{Z}_2).\] In particular, if \([T_r]=[\llbracket\Sigma\rrbracket],\) then \([(f_t)_\#T_r]=[\llbracket\Sigma\rrbracket].\)
Proof. Since \(X\in \mathfrak X^\partial(N)\), its flow preserves the ambient boundary, i.e. \(\psi_t(\partial N)\subset \partial N.\) Also, by construction of the defining
function \(\tau\) in section 3.1, we have \[\nabla\tau\in T\partial N
\qquad\text{on }\partial N,\] and the vector field \[Z=\frac{\nabla\tau}{|\nabla\tau|^2}\qquad\text{tangent to \(\partial N\) along \(\partial N\).}\] Therefore the flow \(F\) of
\(Z\) preserves \(\partial N\), and so does the retraction \(P_r\), i.e. \(P_r(\partial N)\subset \partial N.\)
Since \(P_r\) maps its domain into \(K_r\), we have \[f_t(K_r)=(P_r\circ\psi_t)(K_r)\subset K_r.\] Furthermore, if \(p\in
A_r=K_r\cap\partial N\), then \(\psi_t(p)\in\partial N,\) and hence \[f_t(p)=P_r(\psi_t(p))\in \partial N.\] Since also \(f_t(p)\in K_r\), it follows
that \(f_t(A_r)\subset A_r.\) Thus \[f_t:(K_r,A_r)\to (K_r,A_r)\] is a map of pairs.
Now \(T_r\in\mathcal{Z}_n(K_r,A_r;\mathbb{Z}_2)\) means \[\operatorname{spt}T_r\subset K_r,
\qquad
\operatorname{spt}\partial T_r\subset A_r.\] Since \(f_t(K_r)\subset K_r\), we get \(\operatorname{spt}(f_t)_\#T_r\subset K_r.\) Also, because push-forward commutes with boundary,
\[\partial (f_t)_\#T_r=(f_t)_\#(\partial T_r).\] Since \(f_t(A_r)\subset A_r\), we obtain \[\operatorname{spt}\partial (f_t)_\#T_r
=
\operatorname{spt}(f_t)_\#(\partial T_r)
\subset A_r.\] Therefore \[(f_t)_\#T_r\in\mathcal{Z}_n(K_r,A_r;\mathbb{Z}_2).\]
It remains to show that \((f_t)_\#T_r\) represents the same relative homology class as \(T_r\). Define \[H:[0,1]\times K_r\to K_r \qquad\text{by
}H(\lambda,p)=P_r(\psi_{\lambda t}(p)).\] Since \(P_r|_{K_r}=\operatorname{id}_{K_r}\), we have \[H(0,p)=P_r(p)=p,\quad H(1,p)=P_r(\psi_t(p))=f_t(p).\] Moreover, because \(\psi_{\lambda t}(\partial N)\subset\partial N\) and \(P_r(\partial N)\subset\partial N,\) we have \[H(\lambda,A_r)\subset A_r
\qquad
0\le\lambda\le1.\] Thus \(H\) is a homotopy of pairs between \[\operatorname{id}_{(K_r,A_r)}\quad \text{and \;\;}f_t:(K_r,A_r)\to(K_r,A_r).\]
By the homotopy formula for currents, there exist \[Q_t\in \mathcal{I}_{n+1}(K_r;\mathbb{Z}_2),
\qquad
R_t\in \mathcal{I}_n(A_r;\mathbb{Z}_2),\] such that \[(f_t)_\#T_r-T_r=\partial Q_t+R_t.\] Indeed, one may take \(Q_t=H_\#\bigl([0,1]\times T_r\bigr),\) and the error term supported
in \(A_r\) is \(R_t=H_\#\bigl([0,1]\times \partial T_r\bigr),\) because \(\operatorname{spt}\partial T_r\subset A_r\) and \(H([0,1]\times A_r)\subset A_r.\)
Therefore \[[(f_t)_\#T_r]=[T_r]
\qquad
\text{in }H_n(K_r,A_r;\mathbb{Z}_2).\] In particular, if \([T_r]=[\llbracket\Sigma\rrbracket],\) then \([(f_t)_\#T_r]=[\llbracket\Sigma\rrbracket].\) ◻
Lemma 7 shows that the retracted variation produces legitimate competitors. To extract a first variation estimate from the minimizing property, we also need to know how the
\(n\)-Jacobian of this retracted variation changes to first order in \(t\). This is the content of the next lemma.
The additional term \(C|X|\) below measures the cost of retracting the varied surface back into \(K_r\). It is uniform in \(r\), which is the crucial
point for applying regularity later.
Lemma 8. Let \(X\in \mathfrak X^\partial(N)\), and let \(\psi_t\) be the flow generated by \(X\). Set \(f_t=P_r\circ\psi_t.\) Then there exists a constant \(C>0\), independent of \(r\), such that for every \[y\in K_r,\qquad S\in
G_n(T_yN),\] one has \[J_n(Df_t(y)|_S)
\le
1+t\operatorname{div}_S X(y)+Ct|X(y)|+o(t),\] where \(o(t)/t\to0\) as \(t\to0\), locally uniformly in \((y,S)\).
Proof. Recall that \(F:\Sigma\times(-r_0,r_0)\to \mathcal{U}\) is defined by \(F(x,s)=\Phi_s(x),\) where \(\Phi_s\) is the flow of \(Z=\frac{\nabla\tau}{|\nabla\tau|^2}.\) The ambient Riemannian metric on \(N\) is denoted by \(g\). We pull it back by \(F\) and
write \[\bar g:=F^*g\] for the induced metric on \(\Sigma\times(-r_0,r_0)\).
For each fixed \(s\), let \[F_s:\Sigma\to \Sigma_s \qquad\text{by }F_s(x)=F(x,s).\] Then the metric induced on \(\Sigma\) from the leaf \(\Sigma_s\) is \(g_s:=F_s^*g.\)
For each \(s\in(-r_0,r_0)\), let \[i_s:\Sigma\to \Sigma\times(-r_0,r_0),
\qquad
i_s(x)=(x,s)\] be the inclusion of the \(s\)-slice. Then \(F_s=F\circ i_s\), Therefore, \[g_s:=F_s^*g
=
(F\circ i_s)^*g
=
i_s^*(F^*g)
=
i_s^*\bar g.\] In other words, \(g_s=F_s^*g=i_s^*\bar g.\) Thus \(g_s\) is precisely the restriction of the pulled-back metric \(\bar g\) to the slice
\(\Sigma\times\{s\}\).
Since \(\partial_sF(x,s)=Z(F(x,s)),\) and since \(Z\perp T\Sigma_s,\) the pulled-back metric \(\bar g\) has no mixed terms. Thus \[\bar g=g_s+\lambda^2 ds^2,\] where \[\lambda(x,s)=|Z(F(x,s))|_g=\frac{1}{|\nabla\tau(F(x,s))|_g}.\] In particular, for every \(s\), \[T_{F(x,s)}N=T_{F(x,s)}\Sigma_s\oplus \mathbb{R} Z(F(x,s))\] is an orthogonal decomposition with respect to the ambient metric \(g\).
We first estimate the Jacobian of the leaf-to-leaf map. For \(s,s'\in(-r_0,r_0)\), define \[\mathcal{L}_{s\to s'}:\Sigma_s\to\Sigma_{s'}\qquad\text{by }\mathcal{L}_{s\to
s'}(F(x,s))=F(x,s').\] Equivalently, \[\mathcal{L}_{s\to s'}=F_{s'}\circ F_s^{-1}.\]
Since \(\bar g=F^*g\) is smooth on the compact set \(\Sigma\times[-r_0,r_0],\) the family \(g_s\) depends smoothly on \(s\). Hence there exists \(C_1>0\) such that for all \(\xi\in T_x\Sigma\) and all \(|s|\le r_0\), \[\left|
\frac{d}{ds}g_s(\xi,\xi)
\right|
\le
C_1 g_s(\xi,\xi).\] Indeed, this follows because \(\partial_s g_s\) is a smooth family of symmetric bilinear forms on a compact set.
By Gronwall’s inequality, \[e^{-C_1|s-s'|}g_s(\xi,\xi)
\le
g_{s'}(\xi,\xi)
\le
e^{C_1|s-s'|}g_s(\xi,\xi).\] Therefore, as quadratic forms on \(T_x\Sigma\), \[g_{s'}\le e^{C_1|s-s'|}g_s.\] Taking \(n\)-dimensional
volume elements, using \(|s-s'|<r_0\) and increasing the constant, we obtain \[\begin{align}
J_n\bigl(dL_{s\to s'}|_{T\Sigma_s}\bigr)
&=
\frac{d\operatorname{vol}_{g_{s'}}}{d\operatorname{vol}_{g_s}} \\
&\le
\left(e^{C_1|s-s'|}\right)^{n/2} \\
&=
e^{\frac{n}{2} C_1|s-s'|} \\
&\le
1+C_2|s-s'|.
\end{align}\]
Next we estimate the Jacobian of \(P_r\). Write a point \(z\in \mathcal{U}\) as \(z=F(x,s).\) Recall that \[P_r(F(x,s))=F(x,q_r(s)), \qquad\text{where }q_r(s)=
\begin{cases}
-r,&s<-r,\\
s,&-r\le s\le r,\\
r,&s>r.
\end{cases}\] At points where \(P_r\) is differentiable (Note that by Rademacher’s theorem, it is differentiable almost everywhere), take an arbitrary vector \(v\in T_zN,\) using the
orthogonal splitting \[T_zN=T_z\Sigma_s\oplus \mathbb{R} Z(z),\] write \[v=v^T+aZ(z),
\qquad
v^T\in T_z\Sigma_s.\] Since the splitting is orthogonal, one has \(|v^T|_g\le |v|_g.\)
If \(|s|\le r\), then \(q_r(s)=s\), so \(P_r\) is the identity near \(z\), and hence \(J_n(DP_r(z)|_S)=1.\)
If \(s>r\) or \(s<-r\), then \(q_r(s)\) is constant in \(s\), and \(DP_r\) kills
the \(Z\)-component. More precisely, \[DP_r(z)(v)=d\mathcal{L}_{s\to q_r(s)}(v^T).\] Let \[\operatorname{pr}_s:T_zN\to T_z\Sigma_s\] be the orthogonal
projection. Then \(v^T=\operatorname{pr}_s(v).\) Thus, for every \(n\)-plane \(S\subset T_zN\), \[J_n(DP_r(z)|_S)
\le
J_n(d\mathcal{L}_{s\to q_r(s)}|_{T\Sigma_s})\,
J_n(\operatorname{pr}_s|_S).\] Since \(\operatorname{pr}_s\) is an orthogonal projection, \(J_n(\operatorname{pr}_s|_S)\le1.\) Using the leaf-to-leaf Jacobian estimate, we obtain
\[J_n(DP_r(z)|_S)
\le
1+C_2|s-q_r(s)|.\]
Define the \(\tau\)-distance from \(z=F(x,s)\) to \(K_r\) by \[d_\tau(z,K_r):=|s-q_r(s)|.\] Then the previous estimate
becomes \[J_n(DP_r(z)|_S)
\le
1+C_2d_\tau(z,K_r).\]
Now let \[z_t=\psi_t(y),
\qquad y\in K_r.\] Since \(y\in K_r\), we have \(|\tau(y)|\le r.\) Taylor expansion gives \[\tau(z_t)=\tau(\psi_t(y))
=
\tau(y)+t\,d\tau_y(X(y))+o(t).\] Since \(d\tau\) is bounded on the fixed compact set \(\{|\tau|\le r_0\}\), there exists \(C_3>0\) such that \[|d\tau_y(X(y))|
\le
C_3|X(y)|.\] Therefore \[|\tau(z_t)-\tau(y)|
\le
C_3t|X(y)|+o(t).\] Since \(\tau(y)\in[-r,r]\), the distance from \(\tau(z_t)\) to the interval \([-r,r]\) is bounded by \(|\tau(z_t)-\tau(y)|.\) Hence \[d_\tau(z_t,K_r)
\le
C_3t|X(y)|+o(t).\] Applying the \(P_r\)-Jacobian estimate at \(z_t=\psi_t(y)\), with the plane \[D\psi_t(y)(S)\subset T_{z_t}N,\] we get \[J_n(DP_r(z_t)|_{D\psi_t(y)(S)})
\le
1+C_4t|X(y)|+o(t).\]
On the other hand, the standard first-order expansion for the flow \(\psi_t\) gives \[J_n(D\psi_t(y)|_S)
=
1+t\operatorname{div}_S X(y)+o(t).\]
Now use the chain rule for Jacobians: \[J_n(Df_t(y)|_S)
=
J_n(D(P_r\circ\psi_t)(y)|_S)\] and \[J_n(D(P_r\circ\psi_t)(y)|_S)
\le
J_n(DP_r(z_t)|_{D\psi_t(y)(S)})
\,
J_n(D\psi_t(y)|_S).\] Substituting the two estimates gives \[J_n(Df_t(y)|_S)
\le
\bigl(1+C_4t|X(y)|+o(t)\bigr)
\bigl(1+t\operatorname{div}_S X(y)+o(t)\bigr).\] Expanding the product, \[J_n(Df_t(y)|_S)
\le
1+t\operatorname{div}_S X(y)+C_4t|X(y)|+o(t).\] After renaming the constant, \[J_n(Df_t(y)|_S)
\le
1+t\operatorname{div}_S X(y)+Ct|X(y)|+o(t).\] This is the desired estimate. ◻
We now combine the homological invariance of the retracted variation with the Jacobian estimate. Since \(T_r\) is minimizing in its relative class, comparing it with \((P_r\circ\psi_t)_\#T_r\) gives the desired first variation bound.
Proposition 7 (Uniform bounded first variation). Let \(T_r\) be the mass-minimizing relative mod \(2\) representative of \([\Sigma]\)
in \((K_r,A_r)\) and let \(V_r=|T_r|.\) Then there exists a constant \(C>0\), independent of \(r\), such that for every
\(X\in\mathfrak X^\partial(N),\) one has \[|\delta V_r(X)|
\le
C\int |X|\,d\|V_r\|.\]
Proof. Let \(\psi_t\) be the flow of \(X\), and \(f_t=P_r\circ\psi_t.\) By lemma 7, the current \((f_t)_\#T_r\) is a valid competitor for \(T_r\). Hence, by the minimizing property of \(T_r\), \(\mathbf{M}(T_r)
\le
\mathbf{M}((f_t)_\#T_r).\)
By the area formula for rectifiable currents, \[\mathbf{M}((f_t)_\#T_r)
\le
\int_{G_n(N)}
J_n(Df_t(y)|_S)\,dV_r(y,S).\] Using the Jacobian estimate, \[J_n(Df_t(y)|_S)
\le
1+t\operatorname{div}_S X(y)
+
Ct|X(y)|
+
o(t),\] we obtain \[\mathbf{M}((f_t)_\#T_r)
\le
\int_{G_n(N)}
\left[
1+t\operatorname{div}_S X(y)
+
Ct|X(y)|
+
o(t)
\right]
\,dV_r(y,S).\] Since \(\mathbf{M}(T_r)=\int_{G_n(N)}1\,dV_r,\) the minimizing inequality gives \[0
\le
t\int_{G_n(N)}\operatorname{div}_S X(y)\,dV_r(y,S)
+
Ct\int |X|\,d\|V_r\|
+
o(t)\|V_r\|(K_r).\]
Since \(\|V_r\|(K_r)=\mathbf{M}(T_r)\le \mathbf{M}(\llbracket\Sigma\rrbracket),\) the quantity \(\|V_r\|(K_r)\) is uniformly bounded in \(r\). Thus, after
dividing by \(t>0\) and letting \(t\to0^+\), we get \[0
\le
\delta V_r(X)
+
C\int |X|\,d\|V_r\|.\] Therefore \[\delta V_r(X)
\ge
-C\int |X|\,d\|V_r\|.\]
Applying the same argument to \(-X\), which also belongs to \(\mathfrak X^\partial(N)\), gives \[\delta V_r(-X)
\ge
-C\int |X|\,d\|V_r\|.\] Since \(\delta V_r(-X)=-\delta V_r(X)\), we obtain \(\delta V_r(X)
\le
C\int |X|\,d\|V_r\|.\) Combining the two inequalities yields \[|\delta V_r(X)|
\le
C\int |X|\,d\|V_r\|.\] ◻
This uniform first variation bound is the analytic input needed for the Allard-type regularity theorems. Together with the multiplicity-one varifold convergence proved in the previous subsection, it allows us to promote weak convergence to graphical
convergence.
We now prove that the weak limits obtained above are actually graphical. The argument is local. At interior points we use Allard’s interior regularity theorem. At boundary points we use the free-boundary Allard–type theorem of Grüter–Jost. The
hypotheses of these theorems are verified using the multiplicity-one convergence from section 3.2 and the first variation bound from section 3.3.
Theorem 8 (Grüter–Jost boundary Allard regularity). Let \(n,k\in\mathbb{N}\), let \(p>n\), and let \(\eta>0\). Then there exist
constants \[\gamma=\gamma(n,k,p)>0,
\qquad
\varepsilon=\varepsilon(n,k,p,\eta)>0\] with the following property.
Let \(\Gamma\subset B_\rho^{n+k}(0)\subset \mathbb{R}^{n+k}\) be a \(C^2\) hypersurface with \(0\in\Gamma\). Let \(\Omega\) be one of the two sides of \(\Gamma\). Assume that the rescaled curvature of \(\Gamma\) satisfies \[\kappa_\rho(\Gamma):=
\rho \sup_{\Gamma\cap B_\rho(0)} |A_\Gamma|
\le \varepsilon^2.\] Let \(V\) be an \(n\)-dimensional rectifiable varifold in \(\overline{\Omega}\cap B_\rho(0)\), and denote \(\mu=\|V\|.\) Assume: \[0\in \operatorname{spt}\mu,\qquad\Theta^n(\mu,x)\ge 1
\quad\mu\text{-a.e.},\qquad\frac{\mu(B_\rho(0))}{\omega_n\rho^n}
\le
\frac{1}{2}(1+\varepsilon),\] and assume that \(V\) has free-boundary first variation with generalized mean curvature \(H\in L^p(\mu)\), namely \[\delta
V(X)
=
\int \operatorname{div}_S X(x)\,dV(x,S)
=
-\int X\cdot H\,d\mu\] for every \[X\in C_c^1(B_\rho(0),\mathbb{R}^{n+k})\] satisfying \[X(x)\in T_x\Gamma
\qquad\text{for every }x\in \Gamma.\] Finally assume the scale-invariant mean-curvature smallness condition \[\rho^{1-\frac{n}{p}}
\left(
\int_{B_\rho(0)} |H|^p\,d\mu
\right)^{1/p}
\le \varepsilon.\]
Then, after a Euclidean isometry, there is an \(n\)-dimensional \(C^{1,\alpha}\) graph \(M_u\) such that \[\operatorname{spt}\mu\cap B_{\gamma\rho}(0)
=
M_u\cap B_{\gamma\rho}(0)\cap \overline{\Omega},\] where \[\alpha=\min\left\{\frac{1}{2},1-\frac{n}{p}\right\}.\] More precisely, one can write \[M_u=\{(y,u(y)):y\in
D_{\gamma\rho}\}\] for some domain \(D_{\gamma\rho}\subset\mathbb{R}^n\), with \[u\in C^{1,\alpha}(D_{\gamma\rho},\mathbb{R}^k),
\qquad
u(0)=0,\] and \[\rho^{-1}\sup_{D_{\gamma\rho}} |u|
+
\sup_{D_{\gamma\rho}} |Du|
+
\rho^\alpha
\sup_{\substack{x,y\in D_{\gamma\rho}\\x\ne y}}
\frac{|Du(x)-Du(y)|}{|x-y|^\alpha}
\le C(n,k,p)\eta.\] Moreover \(M_u\) meets \(\Gamma\) orthogonally in the free-boundary sense: \[\nu_\Gamma(x)\in T_xM_u
\qquad
\text{for every }x\in M_u\cap\Gamma\cap B_{\gamma\rho}(0).\]
At interior points, we apply Allard’s interior regularity theorem [11], see also [23].
Theorem 9 (Allard regularity theorem). Let \(n,k\in\mathbb{N}\), \(p>n\), and let \(\eta>0\). Then there exist constants \[\varepsilon=\varepsilon(n,k,p,\eta)>0,
\qquad
\gamma=\gamma(n,k,p)>0,\] with the following property.
Let \(V=v(M,\theta)\) be a rectifiable \(n\)-varifold in \(B_\rho^{n+k}(0)\subset \mathbb{R}^{n+k},\) with weight measure \(\mu=\|V\|.\) Assume that \[0\in \operatorname{spt}\mu,
\qquad
\theta\ge 1 \quad \mu\text{-a.e.},\] and that the density ratio satisfies \[\frac{\mu(B_\rho(0))}{\omega_n\rho^n}\le 1+\varepsilon.\] Assume moreover that \(V\) has generalized mean
curvature \(H\in L^p(\mu)\) in \(B_\rho(0)\), namely \[\int \operatorname{div}_{S}X(x)\,dV(x,S)
=
-\int X\cdot H\,d\mu\] for every \[X\in C_c^1(B_\rho(0),\mathbb{R}^{n+k}),\] and that \[\rho^{1-\frac{n}{p}}
\left(
\int_{B_\rho(0)} |H|^p\,d\mu
\right)^{1/p}
\le \varepsilon.\] Then, after applying an isometry of \(\mathbb{R}^{n+k}\), there exists a function \[u:B_{\gamma\rho}^n(0)\to \mathbb{R}^k\] of class \(C^{1,\alpha}\), where \(\alpha=1-\frac{n}{p},\) such that \[u(0)=0\qquad\text{and}\qquad\operatorname{spt}\mu\cap B_{\gamma\rho}(0)
=
\operatorname{graph}u\cap B_{\gamma\rho}(0).\] Moreover, \[\rho^{-1}\sup_{B_{\gamma\rho}^n(0)}|u|
+
\sup_{B_{\gamma\rho}^n(0)}|Du|
+
\rho^\alpha
[Du]_{C^{0,\alpha}(B_{\gamma\rho}^n(0))}
\le C\eta,\] where \(C=C(n,k,p)>0\).
Remark 10. In the application below, the hypotheses of the two regularity theorems are verified as follows. The density assumption comes from the multiplicity one convergence shown in section 3.2. The mass
ratio bounds are obtained by choosing the radius of the coordinate ball sufficiently small, since \(\Sigma\) is a smooth hypersurface and hence is close to a plane, or to a half-plane at boundary points, at small scales.
Finally, the required first variation bound is exactly the uniform estimate proved in proposition 7; after rescaling to a sufficiently small ball, this gives the
scale-invariant smallness condition required in Allard’s theorem and in the Grüter–Jost theorem.
We now apply the preceding regularity theorems to the minimizers \(T_r\). The main conclusion is that, once \(r\) is sufficiently small, the support of \(T_r\) is not only close to \(\Sigma\) as a varifold, but also is a single free-boundary graph over \(\Sigma\).
Proposition 11 (Graphical convergence of the minimizers). Given \((N^{n+1},\partial N,g)\), \(\Sigma^n\subset N\), \(\tau\), \(Z\), \(F\), \(K_r\), \(A_r\) defined as in section 1. Let \(T_r\in \mathcal{Z}_n(K_r,A_r;\mathbb{Z}_2)\) be a minimizer of mass in the relative homology class \[[T_r]=[\llbracket\Sigma\rrbracket]\in
H_n(K_r,A_r;\mathbb{Z}_2),\] and set \(V_r=|T_r|.\)
Assume that, as \(r\to0\), \(V_r\to |\Sigma|\) as varifolds, with multiplicity one, and that there exists a constant \(C_0>0\), independent of
\(r\), such that for every vector field \(X\in \mathfrak X^\partial(N),\) one has \[|\delta V_r(X)|
\le
C_0\int |X|\,d\|V_r\|.\] Then, after decreasing \(r_0\) if necessary, for every \(0<r<r_0\) there exists a function \[u_r:\Sigma\to (-r,r)\]
such that \[\operatorname{spt}\|V_r\|=\operatorname{spt}T_r
=
\Gamma_{u_r}
:=
\{F(x,u_r(x)):x\in\Sigma\}.\] Moreover \[T_r=\llbracket \Gamma_{u_r}\rrbracket
\quadin \mathcal{Z}_n(K_r,A_r;\mathbb{Z}_2),\] and, for every \(\alpha\in(0,1)\), \(u_r\to0\) locally in \(C^{1,\alpha}\) in the interior of \(\Sigma\), and locally in the free boundary \(C^{1,\alpha}\) sense near \(\partial\Sigma\). In particular, \[u_r\to0
\quadin C^1(\Sigma).\]
Proof. We write the proof in several steps.
Step 1. Interior flatness.
The purpose of this step is to verify the flatness and multiplicity-one plane approximation hypotheses in the interior Allard regularity theorem.
Let \[p\in\operatorname{int}\Sigma,
\qquad
P_p=T_p\Sigma.\] Let \[\varepsilon_{\mathrm A}>0,
\qquad
\gamma_{\mathrm A}\in(0,1)\] be the smallness threshold and the shrinking factor corresponds to \(\eta_A\) in the interior Allard regularity theorem.
Choose \(R_p>0\) sufficiently small such that \(B_{2R_p}(p)\Subset N^\circ,\)\(\Sigma\cap B_{2R_p}(p)\) is \(C^2\)-close to \(P_p\), and \(C_0R_p<\varepsilon_{\mathrm A}.\)
By the identity \((\pi)_\#T_r=\llbracket\Sigma\rrbracket\) shown in lemma 2, we have \(\pi(\operatorname{spt}T_r)=\Sigma.\) Hence we may choose \(p_r\in\operatorname{spt}\|V_r\|\cap \pi^{-1}(p)\) with \(d(p_r,p)\le Cr.\) Thus, for \(r\) sufficiently small, \[d(p_r,p)<\frac{1}{10}\gamma_{\mathrm A}R_p\qquad\text{and}\qquad B_{R_p}(p_r)\subset B_{2R_p}(p).\]
We claim that \(V_r\) satisfies the flatness and multiplicity-one hypotheses of the interior Allard theorem in \(B_{R_p}(p_r)\), relative to the affine plane through \(p_r\) parallel to \(P_p\). Indeed, since \(\Sigma\) is smooth and \(p_r\to p\), the hypersurface \(\Sigma\) is arbitrarily close to this affine plane in \(B_{R_p}(p_r)\), after first choosing \(R_p\) small. Since \(V_r\to|\Sigma|\) as varifolds with multiplicity one convergence, the mass ratio of \(V_r\) in \(B_{R_p}(p_r)\) is as small as required for all sufficiently small
\(r\), while the density and support condition naturally holds.
Step 2. Interior first variation smallness after rescaling.
The purpose of this step is to match the first variation hypothesis in the interior Allard regularity Theorem 9.
By assumption, for all compactly supported vector fields \(X\) in \(B_{2R_p}(p)\), we have \[|\delta V_r(X)|
\le
C_0\int |X|\,d\|V_r\|.\] Since \(B_{2R_p}(p)\Subset N^\circ,\) every compactly supported vector field in the ball is automatically admissible.
Rescale the ball \(B_{R_p}(p_r)\) to a unit ball by \[\eta_{p_r,R_p}(y)=\frac{y-p_r}{R_p}.\] Let \(\widetilde{V}_r=(\eta_{p_r,R_p})_\#V_r.\) Under this
scaling, the first variation bound becomes \[|\delta \widetilde{V}_r(Y)|
\le
C_0R_p\int |Y|\,d\|\widetilde{V}_r\|\] for vector fields \(Y\) on the rescaled ball. Therefore the scale-invariant mean curvature quantity is bounded by \(C_0R_p.\) After decreasing
\(R_p\) if necessary, we may assume \(C_0R_p<\varepsilon_{\mathrm A}.\) Thus the first variation smallness condition in the interior Allard theorem is satisfied.
Step 3. Application of the interior Allard theorem.
By steps 1 and 2, the interior Allard regularity theorem applies to \(V_r\) in \(B_{R_p}(p_r)\). Therefore \[\operatorname{spt}\|V_r\|\cap B_{\gamma_{\mathrm
A}R_p}(p_r)\] is a \(C^{1,\alpha_{\mathrm A}}\) graph.
Since \(d(p_r,p)<\frac{1}{10}\gamma_{\mathrm A}R_p,\) we have \[B_{\frac{1}{2}\gamma_{\mathrm A}R_p}(p)
\subset
B_{\gamma_{\mathrm A}R_p}(p_r).\] Consequently \[\operatorname{spt}\|V_r\|\cap B_{\frac{1}{2}\gamma_{\mathrm A}R_p}(p)\] is a \(C^{1,\alpha_{\mathrm A}}\) graph which is \(C^1\)-close to \(\Sigma\), after first choosing \(R_p\) small and then taking \(r\) sufficiently small.
Step 4. Boundary flatness.
The purpose of this step is to verify the support, boundary-curvature, density, and mass-ratio hypotheses in the Gruter–Jost boundary regularity Theorem 8.
Let \(p\in\partial\Sigma ,\) let \(\eta_{\mathrm{GJ}}>0\) be a small graphical parameter, and let \[\varepsilon_{\mathrm{GJ}}>0,
\qquad
\gamma_{\mathrm{GJ}}\in(0,1)\] be the constants in the Gruter–Jost boundary regularity theorem 8, corresponding to \(\eta_{\mathrm{GJ}}\). We also fix an
exponent \(q>n\) in the mean-curvature assumption of theorem 8.
With same argument in step 1, we may choose \[p_r\in \operatorname{spt}\|V_r\|\cap \pi^{-1}(p),\] where whole fibre \(\pi^{-1}(p)=\{F(p,s): |s|<r_0\}\) is contained in \(\partial N\). Therefore \[p_r\in \operatorname{spt}\|V_r\|\cap\partial N .\] Moreover, as in step 1, one has \(d(p_r,p)\le Cr.\)
Choose \(R_p>0\) sufficiently small such that the following conditions hold:
The Fermi coordinate charts centered at points near \(p\) are defined in balls of radius \(2R_p\).
The rescaled curvature of \(\partial N\) is small: \(R_p\sup_{\partial N\cap B_{2R_p}(p)} |A_{\partial N}|\le\varepsilon_{\mathrm{GJ}}^2.\)
\(\Sigma\cap B_{2R_p}(p)\) is \(C^2\)-close to its tangent half-plane at \(p\).
\(C_0R_p<\varepsilon_{\mathrm{GJ}}.\)
For \(r\) sufficiently small, since \(d(p_r,p)\le Cr\), we have \[d(p_r,p)<\frac{1}{10}\gamma_{\mathrm{GJ}}R_p\] and \[B_{R_p}(p_r)\subset B_{2R_p}(p).\] In the Fermi coordinate chart centered at \(p_r\), the point \(p_r\) is sent to the origin, \(N\) is represented as one side \(\Omega_r\) of the boundary hypersurface \(\Gamma_r^\partial\), and \(0\in \Gamma_r^\partial .\)
The curvature condition in theorem 8 is satisfied because \[R_p\sup_{\Gamma_r^\partial\cap B_{R_p}(0)} |A_{\Gamma_r^\partial}|
\le
\varepsilon_{\mathrm{GJ}}^2,\] by our choice of \(R_p\).
We now check the support and density assumptions. Since \(p_r\in\operatorname{spt}\|V_r\|,\) we have, in the Fermi coordinates centered at \(p_r\), \(0\in\operatorname{spt}\|V_r\|.\)
Since \(V_r=|T_r|\) is an integral varifold, we have \(\Theta^n(\|V_r\|,x)\ge1
\text{ for }
\|V_r\|-a.e.\)
It remains to verify the boundary mass-ratio condition. Since \(p_r\to p\) and \(\Sigma\) is a smooth free-boundary hypersurface, after choosing \(R_p\)
sufficiently small we have \[\frac{
\mathcal{H}^n(\Sigma\cap B_{R_p}(p_r))
}{
\omega_n R_p^n
}
\le
\frac{1}{2}\left(1+\frac{\varepsilon_{\mathrm{GJ}}}{4}\right)\] for all sufficiently small \(r\). Here the factor \(1/2\) comes from the fact that \(p\in\partial\Sigma\), so \(\Sigma\) has half-plane density at \(p\).
Since \(V_r\to|\Sigma|\) as varifolds with multiplicity one, and since \(p_r\to p\), after choosing \(R_p\) to be a regular radius for the measure \(\mathcal{H}^n\llcorner\Sigma\), we obtain \[\frac{
\|V_r\|(B_{R_p}(p_r))
}{
\omega_n R_p^n
}
\le
\frac{1}{2}(1+\varepsilon_{\mathrm{GJ}})\] for all sufficiently small \(r\). Thus the mass-ratio hypothesis in the Grúter–Jost theorem is satisfied.
Step 5. Boundary first variation smallness after rescaling.
The purpose of this step is to match the first variation hypothesis in the Grüter–Jost regularity Theorem 8.
The vector fields used in the Grüter–Jost regularity Theorem 8 are tangent to the boundary hyperplane. In our notation these are exactly the local representatives of vector fields \(X\in \mathfrak X^\partial(N).\) By assumption, \[|\delta V_r(X)|
\le
C_0\int |X|\,d\|V_r\|\] for all such admissible vector fields.
Same as in step 2, after rescaling \(B_{R_p}^+\) to a unit half-ball, this estimate becomes \[|\delta \widetilde{V}_r(Y)|
\le
C_0R_p\int |Y|\,d\|\widetilde{V}_r\|.\] Therefore the scale-invariant first variation quantity is bounded by \(C_0R_p.\) After decreasing \(R_p\) if necessary, we may assume \(C_0R_p<\varepsilon_{\mathrm{GJ}}.\) Thus the first variation smallness hypothesis in the Gruter–Jost theorem is satisfied.
Step 6. Application of the Gruter–Jost boundary regularity theorem.
By Steps 4 and 5, all hypotheses of the Gruter–Jost theorem 8 are satisfied for \(V_r\) in the Fermi coordinate half-ball centered at \(p_r.\) Therefore \[\Gamma_r\cap B_{\gamma_{\mathrm{GJ}}R_p}(p_r),
\qquad
\Gamma_r:=\operatorname{spt}\|V_r\|,\] is a \(C^{1,\alpha_{\mathrm{GJ}}}\) free-boundary hypersurface. Moreover it meets \(\partial N\) orthogonally, and the \(C^1\)-norm of the local graph given by theorem 8 is bounded by \(C(n,q)\eta_{\mathrm{GJ}}.\)
Since \(d(p_r,p)<\frac{1}{10}\gamma_{\mathrm{GJ}}R_p,\) we have \[B_{\frac{1}{2}\gamma_{\mathrm{GJ}}R_p}(p)\cap N
\subset
B_{\gamma_{\mathrm{GJ}}R_p}(p_r)\cap N.\] Consequently \[\Gamma_r\cap B_{\frac{1}{2}\gamma_{\mathrm{GJ}}R_p}(p)\] is a \(C^{1,\alpha_{\mathrm{GJ}}}\) free-boundary hypersurface which
is \(C^1\)-close to \(\Sigma\), after first choosing \(R_p\) small and then taking \(r\) sufficiently small.
The regularity theorems give only local graphs over tangent planes. We must still show that these local graphs patch together into one global graph over \(\Sigma\), rather than several sheets. This is where multiplicity
one and the mass convergence are used again.
Step 7. From local regularity to a global graph over \(\Sigma\).
The purpose of this step is to use the local \(C^{1,\alpha}\) regularity obtained in Steps 3 and 6 to prove that \(\operatorname{spt}\|V_r\|\) is a single global graph over \(\Sigma\). We do not glue the local planar graph functions given by the Allard and Gruter–Jost theorems. Those functions are only auxiliary. Instead, we use the tubular projection \(\pi:\mathcal{U}\to\Sigma.\)
Set \[\Gamma_r:=\operatorname{spt}\|V_r\|=\operatorname{spt}T_r.\] By Steps 3 and 6, for every \(p\in\Sigma\), there exists a neighborhood \(W_p\) of
\(p\) such that, for \(r\) sufficiently small, \[\Gamma_r\cap W_p\] is a \(C^{1,\alpha}\) hypersurface, free-boundary near
\(\partial\Sigma\), and is \(C^1\)-close to \(\Sigma\). More precisely, the tangent spaces satisfy \[\operatorname{dist}_{\mathcal{G}}
\bigl(
T_y\Gamma_r,
T_{\pi(y)}\Sigma
\bigr)\] arbitrarily small for \(y\in\Gamma_r\cap W_p\), after first choosing the regularity scale \(R_p\) sufficiently small and then taking \(r\)
sufficiently small.
Since \(\Sigma\) is compact, choose finitely many points \(p_1,\ldots,p_J\in\Sigma\) such that \(\Sigma\subset \bigcup_{i=1}^J W_{p_i}.\) Since \(\Gamma_r\subset K_r=\{|\tau|\le r\}\) and \(K_r\) converges to \(\Sigma\) as \(r\to0\), after taking \(r\) smaller if necessary we have \[\Gamma_r\subset \bigcup_{i=1}^J W_{p_i}.\] Thus the above local regularity covers the whole \(\Gamma_r\).
We now show that \[\pi|_{\Gamma_r}:\Gamma_r\to\Sigma\] is a local diffeomorphism. Indeed, in the tubular coordinates \(F(x,s)\), the kernel of \(D\pi\)
is precisely the line generated by \(Z\): \[\ker D\pi_y=\operatorname{span}\{Z(y)\}.\] Since \(T_y\Gamma_r\) is \(C^1\)-close to \(T_{\pi(y)}\Sigma\), and \(T_{\pi(y)}\Sigma\) is transverse to the \(Z\)-direction, \(T_y\Gamma_r\) is also transverse to the \(Z\)-direction. Hence \[D(\pi|_{\Gamma_r})_y:
T_y\Gamma_r\to T_{\pi(y)}\Sigma\] is an isomorphism for every \(y\in\Gamma_r\). Near the free boundary this is understood in the sense of manifolds with boundary. Therefore \(\pi|_{\Gamma_r}\) is a local diffeomorphism.
Next we prove that \(\pi|_{\Gamma_r}\) is onto. By Lemma 2, \((\pi)_\#T_r=\llbracket\Sigma\rrbracket.\)
Since \[\operatorname{spt}\bigl((\pi)_\#T_r\bigr)
\subset
\pi(\operatorname{spt}T_r)
=
\pi(\Gamma_r)\qquad\text{and}\qquad\operatorname{spt}\llbracket\Sigma\rrbracket=\Sigma,\] we obtain \(\Sigma\subset \pi(\Gamma_r).\) The reverse inclusion follows from the definition of \(\pi\). Thus \(\pi(\Gamma_r)=\Sigma.\)
Consequently, \(\pi|_{\Gamma_r}:\Gamma_r\to\Sigma\) is a proper surjective local diffeomorphism. Hence it is a finite-sheeted covering map. On each connected component \(\Sigma_j\) of
\(\Sigma\), let \(m_j\in\mathbb{N}\) be the number of sheets over \(\Sigma_j\).
We claim that \[m_j=1
\qquad
\text{for every connected component }\Sigma_j.\] Suppose, to the contrary, that \(m_j\ge2\) for some \(j\). Since each sheet is \(C^1\)-close to \(\Sigma_j\), the area of each sheet is \[(1+o(1))\mathcal{H}^n(\Sigma_j)\qquad\text{as \(r\to0\).}\] Hence \[\mathbf{M}(T_r)
=
\|V_r\|(N)
\ge
m_j(1+o(1))\mathcal{H}^n(\Sigma_j)
+
\sum_{\ell\ne j}(1+o(1))\mathcal{H}^n(\Sigma_\ell).\] If \(m_j\ge2\), then for all sufficiently small \(r\) this is strictly larger than \(\mathcal{H}^n(\Sigma).\) This contradicts the multiplicity-one varifold convergence \(V_r\to|\Sigma|,\) which implies \(\|V_r\|(N)\to\mathcal{H}^n(\Sigma).\)
Therefore \(m_j=1\) for every \(j\). Hence \(\pi|_{\Gamma_r}:\Gamma_r\to\Sigma\) is a one-sheeted covering map, and therefore a global diffeomorphism.
Define \[\sigma_r:\Sigma\to\Gamma_r \qquad\text{as }\sigma_r=(\pi|_{\Gamma_r})^{-1}.\] Define \[u_r:\Sigma\to(-r,r) \qquad\text{by }u_r(x):=\tau(\sigma_r(x)).\] Since \(\pi(\sigma_r(x))=x,\) the point \(\sigma_r(x)\) lies on the \(Z\)-flow line through \(x\). Hence there exists a unique \(s\in(-r,r)\) such that \[\sigma_r(x)=F(x,s).\] Since \(\tau(F(x,s))=s,\) this value of \(s\) is exactly \(u_r(x)\). Therefore \[\sigma_r(x)=F(x,u_r(x)).\] It follows that \[\Gamma_r
=
\{F(x,u_r(x)):x\in\Sigma\}.\] Moreover \(u_r\) is locally \(C^{1,\alpha_\text{A}}\) in the interior and locally free-boundary \(C^{1,\alpha_\text{GJ}}\) near \(\partial\Sigma\).
Step 8. Identification of the current and \(C^1\)-convergence to zero.
The purpose of this step is to identify \(T_r\) with the \(\mathbb{Z}_2\)-current induced by the graph \(\Gamma_{u_r}\), and to prove that the graph
functions converge to zero in \(C^1\).
By Step 7, \[\Gamma_r
=
\Gamma_{u_r}
=
\{F(x,u_r(x)):x\in\Sigma\}.\] Since \(T_r\) is a \(\mathbb{Z}_2\)-integral current supported on the smooth hypersurface \(\Gamma_{u_r}\), it is
represented by a \(\mathbb{Z}_2\)-multiplicity function on \(\Gamma_{u_r}\). Because \(\pi|_{\Gamma_{u_r}}:\Gamma_{u_r}\to\Sigma\) is a diffeomorphism and
\((\pi)_\#T_r=\llbracket\Sigma\rrbracket,\) the multiplicity must be equal to \(1\) almost everywhere on \(\Gamma_{u_r}\). Hence \(T_r=\llbracket\Gamma_{u_r}\rrbracket\) as a \(\mathbb{Z}_2\)-current.
It remains to prove \[u_r\to0
\qquad
\text{in }C^1(\Sigma).\] Let \(\eta>0\). In Steps 1–6, choose the regularity scales \(R_{p_i}\) in the finite cover sufficiently small so that \(\Sigma\) is \(C^2\)-close to its tangent plane or tangent half-plane in each chart, and so that the scale-invariant first variation terms are smaller than \(\varepsilon\). Then take \(r\) sufficiently small so that the varifold convergence \(V_r\to|\Sigma|\) makes all the required mass-ratio and density hypotheses
hold in the finitely many chosen charts.
The Allard and Gruter–Jost estimates then imply that, in each chart, \(\Gamma_r\) is \(C^1\)-close to \(\Sigma\), with \(C^1\)-distance bounded by \(C\eta\). Since the cover is finite, we get \[\|u_r\|_{C^1(\Sigma)}\le C\eta\] for all sufficiently small \(r\). Since \(\eta>0\) is arbitrary, this gives \[u_r\to0
\qquad
\text{in }C^1(\Sigma).\] This completes the proof. ◻
At this point the problem has been reduced to the classical graphical setting. The minimizer \(T_r\) is represented by a free-boundary graph \(\Gamma_{u_r}\) over \(\Sigma\), with \(u_r\to0\) in \(C^1\). We can therefore use the strict stability of \(\Sigma\), which gives strict local
minimality for the area functional among small free-boundary graphs.
Proposition 12. Let \(\Sigma^n\subset (N^{n+1},g)\) be a smooth compact two-sided properly embedded free boundary minimal hypersurface. Assume that \(\Sigma\) is
strictly stable, namely its free-boundary second variation form \[Q(\phi,\phi)
=
\int_\Sigma
\left(
|\nabla^\Sigma\phi|^2
-
\bigl(|A_\Sigma|^2+\operatorname{Ric}_N(\nu,\nu)\bigr)\phi^2
\right)
d\mu_\Sigma
-
\int_{\partial\Sigma}
h^{\partial N}(\nu,\nu)\phi^2
d\sigma\] is positive definite.
Let \(\tau,F,K_r,A_r,T_r,V_r\) be same as in above sections. Suppose that, for \(r\to0\), \[T_r=\llbracket\Gamma_{u_r}\rrbracket,
\qquad
\Gamma_{u_r}=\{F(x,u_r(x)):x\in\Sigma\},
\qquad
u_r\to0
\quad\text{in }
C^1(\Sigma).\] Then, for all sufficiently small \(r\), \(T_r=\llbracket\Sigma\rrbracket.\) Consequently, \(\mathbf{M}(T)\ge
\mathbf{M}(\llbracket\Sigma\rrbracket)\) for every \(T\in\mathcal{Z}_n(K_r,A_r;\mathbb{Z}_2)\) satisfying \[[T]=[\llbracket\Sigma\rrbracket]
\quad
\operatorname{in}
\quad
H_n(K_r,A_r;\mathbb{Z}_2).\] Thus \(\Sigma\) is locally area-minimizing in its relative homology class.
Proof. First introduce the graph area functional \[\mathcal{A}(u)=\mathcal{H}^n(\Gamma_u),
\qquad
\Gamma_u=\{F(x,u(x)):x\in\Sigma\}.\] Since \(F(\partial\Sigma,s)\subset\partial N\), each \(\Gamma_u\) is an admissible graph competitor for \(\|u\|_{C^1}\) small.
Since \(\Sigma\) is a free boundary minimal hypersurface, \[D\mathcal{A}(0)[u]=0.\] Moreover, \[D^2\mathcal{A}(0)[u,u]=Q(au,au),\] where \[Z|_\Sigma=a\nu_{\Sigma},
\qquad
a=\langle Z,\nu_\Sigma\rangle>0.\] Since \(a\) is smooth and strictly positive, strict stability implies that there exists \(c_0>0\) such that \[D^2\mathcal{A}(0)[u,u]
\ge
c_0\|u\|_{H^1(\Sigma)}^2.\]
The area integrand for \(\Gamma_u\) is a smooth function of \(u\) and \(\nabla u\). Hence, for \(\|u\|_{C^1}\)
sufficiently small, Taylor expansion gives \[\mathcal{A}(u)-\mathcal{A}(0)
=
\frac{1}{2}D^2\mathcal{A}(0)[u,u]+R(u),\] with \[|R(u)|
\le
C\|u\|_{C^1}\|u\|_{H^1(\Sigma)}^2.\] Therefore, after decreasing the \(C^1\)-neighborhood of \(0\), we obtain \[\mathcal{A}(u)-\mathcal{A}(0)
\ge
c_1\|u\|_{H^1(\Sigma)}^2\] for some \(c_1>0\).
Now apply this to \(u=u_r\). Since \[u_r\to0
\quad
C^1(\Sigma),\] for \(r\) sufficiently small the above estimate applies. Hence \[\mathbf{M}(T_r)
=
\mathcal{H}^n(\Gamma_{u_r})
=
\mathcal{A}(u_r)
\ge
\mathcal{A}(0)
+
c_1\|u_r\|_{H^1(\Sigma)}^2.\] Since \[\mathcal{A}(0)=\mathcal{H}^n(\Sigma)=\mathbf{M}(\llbracket\Sigma\rrbracket),\] we have \[\mathbf{M}(T_r)
\ge
\mathbf{M}(\llbracket\Sigma\rrbracket)
+
c_1\|u_r\|_{H^1(\Sigma)}^2.\]
On the other hand, \(T_r\) minimizes mass in the relative homology class of \(\llbracket\Sigma\rrbracket\) in \(K_r\), and \(\llbracket\Sigma\rrbracket\) is itself an admissible competitor. Therefore \(\mathbf{M}(T_r)\le \mathbf{M}(\llbracket\Sigma\rrbracket).\)
Combining the two inequalities gives \(c_1\|u_r\|_{H^1(\Sigma)}^2\le0.\) Thus \[u_r\equiv0 \qquad \text{for sufficiently small }r.\] Hence \[\Gamma_{u_r}=\Sigma,
\qquad
T_r=\llbracket\Sigma\rrbracket.\]
Finally, since \(T_r\) was chosen as a mass minimizer among all currents \(T\in\mathcal{Z}_n(K_r,A_r;\mathbb{Z}_2)\) with \([T]=[\llbracket\Sigma\rrbracket],\) we obtain, for every such \(T\), \[\mathbf{M}(T)\ge \mathbf{M}(T_r)=\mathbf{M}(\llbracket\Sigma\rrbracket).\] This proves
that \(\Sigma\) is locally area-minimizing in its relative homology class. This also completes the proof of Theorem 4. ◻
Remark 13 (On the higher codimension case). The codimension-one assumption is used in an essential way in our construction of the tubular neighborhood and in the Jacobian estimate for the retraction. Indeed, in the hypersurface case we
use a scalar defining function \(\tau:U\to\mathbb{R}\) and the vector field \(Z=\frac{\nabla\tau}{|\nabla\tau|^2}.\) The flow of \(Z\) gives an orthogonal
foliation by level sets of \(\tau\), and this orthogonality is crucial in proving the retraction estimate without producing an unwanted zeroth-order \(O(r)\) error. In higher codimension, a
single scalar defining function is no longer available, and an arbitrary adapted normal-bundle parametrization generally does not give such an orthogonal flow. Therefore a direct extension to arbitrary higher codimension meets a genuine obstruction at the
level of the first variation estimate.
However, if the normal bundle of \(\Sigma^n\subset N^{n+q}\) is trivial, then one may expect the argument to extend by using a global vector-valued defining map \[\tau=(\tau^1,\ldots,\tau^q):U\to\mathbb{R}^q\] satisfying \(\Sigma=\tau^{-1}(0),
\;
\operatorname{rank}d\tau=q,\) and in the free-boundary setting, \[\partial_{\mu_{\partial N}}\tau^\alpha=0
\quad
\operatorname{on}\;U\cap\partial N,
\qquad
\alpha=1,\ldots,q.\] Then the radial function \(\rho=|\tau|\) is smooth on \(U\setminus\Sigma\), satisfies \[\partial_{\mu_{\partial N}}\rho=0
\quad
\operatorname{on}\;(U\cap\partial N)\setminus\partial\Sigma,\] and hence \[\nabla\rho\in T\partial N
\quad
\operatorname{on}\;\partial N\setminus\partial\Sigma.\] Thus the vector field \(Y=\frac{\nabla\rho}{|\nabla\rho|^2}\) is tangent to \(\partial N\) along the boundary and satisfies
\(d\rho(Y)=1.\) Its flow gives a boundary-preserving radial retraction onto the sets \(K_r=\{\rho\le r\}.\)
The key estimate needed for the higher codimension argument is the one-sided metric estimate along the level sets \(\Lambda_s=\{\rho=s\}.\) Although \(\rho\) is not smooth on \(\Sigma\), the same strategy should apply, provided that the following retraction estimate is established: \[\frac{d}{ds}g_s(W,W)\ge -C g_s(W,W),
\qquad
W\in T\Lambda_s,\] where \(g_s\) is the metric induced on \(\Lambda_s\). By Gronwall’s inequality, this yields \[J_m(DP_r(z)|_S)
\le
1+C(\rho(z)-r)_+\] for every \(m\)-plane \(S\subset T_zN\), with \(C\) independent of \(r\). This is precisely the
estimate needed to recover the uniform first variation bound. Once this retraction estimate is established, the remaining parts of the proof, namely the relative homology argument, the mass and varifold convergence, the regularity step, and the final
strict stability argument, should extend with only notational changes. We do not pursue this higher codimension extension here.
We end this section by recording a simple consequence of the local minimizing property.
Corollary 2. Fix \(0<r<r_0\), \(K_r=\{|\tau|\le r\},\)\(A_r=K_r\cap\partial N\) as in Theorem 4. Let \[\mathcal{C}_r
:=
\left\{
T\in\mathcal{Z}_n(K_r,A_r;\mathbb{Z}_2):
[T]=[\llbracket\Sigma\rrbracket]
\text{ in }H_n(K_r,A_r;\mathbb{Z}_2)
\right\}.\] Assume that \(\llbracket\Sigma\rrbracket\) is the unique mass minimizer in \(\mathcal{C}_r\).
Let \(\mathcal{O}\subset \mathcal{C}_r\) be an open neighborhood of \(\llbracket\Sigma\rrbracket\) with respect to the \(\mathcal{F}_{rel}\)-topology.
Then there exists \(\delta=\delta(r,\mathcal{O})>0,\) such that for every \(T\in\mathcal{C}_r\setminus\mathcal{O},\) one has \[\mathbf{M}(T)
\ge
\mathbf{M}(\llbracket\Sigma\rrbracket)+\delta.\]
Proof. Suppose by contradiction that there exists a sequence \(T_j\in\mathcal{C}_r\setminus\mathcal{O}\) such that \(\mathbf{M}(T_j)\to \mathbf{M}(\llbracket\Sigma\rrbracket).\)
Since \(T_j\in\mathcal{C}_r,\) and \(\llbracket\Sigma\rrbracket\) is a minimizer in \(\mathcal{C}_r\), we have \(\mathbf{M}(T_j)\ge
\mathbf{M}(\llbracket\Sigma\rrbracket).\) Therefore \[\mathbf{M}(T_j)\downarrow \mathbf{M}(\llbracket\Sigma\rrbracket)\] after passing to a subsequence if necessary.
Because \[\operatorname{spt}T_j\subset K_r\qquad\text{and}\qquad\sup_j\mathbf{M}(T_j)<C,\quad\text{for some }C>0\text{ and sufficiently large }j\] the Federer–Fleming compactness theorem gives a subsequence, still
denoted by \(T_j\), and a current \(T_\infty\in\mathcal{Z}_n(K_r,A_r;\mathbb{Z}_2)\) such that \[T_j\to T_\infty \qquad\text{in the relative flat topology.
}\] Since relative homology classes are closed under relative flat convergence, and since for each \(j\), \([T_j]=[\llbracket\Sigma\rrbracket],\) we have \([T_\infty]=[\llbracket\Sigma\rrbracket].\) Thus \(T_\infty\in\mathcal{C}_r.\)
By lower semicontinuity of mass under flat convergence, \[\mathbf{M}(T_\infty)
\le
\liminf_{j\to\infty}\mathbf{M}(T_j)
=
\mathbf{M}(\llbracket\Sigma\rrbracket).\] On the other hand, since \(T_\infty\in\mathcal{C}_r\) and \(\llbracket\Sigma\rrbracket\) is a minimizer in \(\mathcal{C}_r\), we have \(\mathbf{M}(T_\infty)
\ge
\mathbf{M}(\llbracket\Sigma\rrbracket).\) Hence \[\mathbf{M}(T_\infty)
=
\mathbf{M}(\llbracket\Sigma\rrbracket).\] By uniqueness of the minimizer, \(T_\infty=\llbracket\Sigma\rrbracket.\)
Since \(\mathcal{O}\) is an open \(\mathcal{F}_{rel}\)-neighborhood of \(\llbracket\Sigma\rrbracket,\) it follows that \(T_j\in\mathcal{O}\) for all sufficiently large \(j\). This contradicts \(T_j\in\mathcal{C}_r\setminus\mathcal{O}.\) ◻
Remark 14. The \(\mathcal{F}_{rel}\) in this corollary could be upgrade to \[\mathbf{F}_{rel}(T,S)=\mathcal{F}_{rel}(T-S)+\mathbf{F}(|T|,|S|)\] by the mass convergence.
Note that the definition of \(\mathbf{F}_{rel}\) is same as in [3]
4 Proof of Theorem 5 and an Application to First
Width↩︎
4.1 From tubular neighborhoods to flat neighborhoods↩︎
The tubular-neighborhood version proved in Section 3 is the natural local statement from the viewpoint of the proof, since the projection, the retraction, and the first-variation estimates are all constructed inside \(K_r\). However, from the viewpoint of min–max theory, the tubular support condition is not the most natural one. Sweepouts in the Almgren–Pitts theory are continuous in the flat or \(\mathbf{F}\)-topology, and a slice which is close to \(\llbracket\Sigma\rrbracket\) in this topology need not have support contained in a prescribed tubular neighborhood of \(\Sigma\).
The goal of this section is to remove this support restriction. We prove that the local minimizing property obtained in Theorem 4 persists in a relative flat neighborhood of \(\llbracket\Sigma\rrbracket\). This strengthening is important because it makes the local theorem compatible with the topology of the free-boundary min–max cycle space in [2], [3], [22]. For similar result, see [5].
The argument follows the exterior-replacement strategy of Marques–Neves [4]. Given a flat-close competitor, we keep it fixed in a smaller neighborhood \(V\Subset K_r\) and minimize outside \(V\). The resulting replacement is stationary with free boundary away from \(V\). The free-boundary monotonicity formula
then prevents the replacement from reaching \(N\setminus K_r\), because otherwise it would carry a definite amount of mass outside \(V\), contradicting flat convergence to \(\Sigma\). Hence the replacement is forced into \(K_r\), where Theorem 4 applies.
We keep the notation from Theorem 4, and write \(\mathcal{Z}_n(N,\partial N;\mathbb{Z}_2)\) for the space of relative \(n\)-cycles in \((N,\partial N)\), and \(\mathcal{F}_{\mathrm{rel}}^N\) for the relative flat distance in the ambient pair \((N,\partial
N)\). We first restate the theorem:
Theorem 15 (Relative flat-neighborhood local minimality). Let \((N^{n+1},g)\) be a compact smooth Riemannian manifold with boundary, and let \(\Sigma^n\subset N\) be a
compact, two-sided, properly embedded, strictly stable free-boundary minimal hypersurface. Then there exists \(\varepsilon>0\) such that the following holds.
If \[S\in\mathcal{Z}_n(N,\partial N;\mathbb{Z}_2),
\qquad
[S]=[\llbracket\Sigma\rrbracket]\in H_n(N,\partial N;\mathbb{Z}_2),\] and \[\mathcal{F}_{\mathrm{rel}}^N(S-\llbracket\Sigma\rrbracket)<\varepsilon,\] then \[\mathbf{M}(S)\ge
\mathbf{M}(\llbracket\Sigma\rrbracket).\] Moreover, if equality holds, then \(S=\llbracket\Sigma\rrbracket\) as a relative \(\mathbb{Z}_2\)-cycle in \((N,\partial N)\).
Proof. Fix \(0<r<r_0\) sufficiently small so that Theorem 4 applies on \(K_r=F(\Sigma\times[-r,r]).\)
Thus \(\llbracket\Sigma\rrbracket\) is the unique mass minimizer in its relative homology class in \((K_r,A_r).\) Choose \(0<\rho<r\), and set \[V:=F(\Sigma\times(-\rho,\rho)).\] Then \(\overline{V}\subset \operatorname{int}K_r\) relative to \(N\).
Suppose by contradiction that the conclusion fails. Then there exists a sequence \[S_j\in \mathcal{Z}_n(N,\partial N;\mathbb{Z}_2)\] such that \[[S_j]=[\llbracket\Sigma\rrbracket]
\quad\text{in }H_n(N,\partial N;\mathbb{Z}_2),\qquad\mathcal{F}_{\mathrm{rel}}^N(S_j-\llbracket\Sigma\rrbracket)\to0,\] but \[S_j\ne \llbracket\Sigma\rrbracket,
\qquad
\mathbf{M}(S_j)\le \mathbf{M}(\llbracket\Sigma\rrbracket).\] By lower semicontinuity of mass under relative flat convergence, \[\mathbf{M}(\llbracket\Sigma\rrbracket)
\le
\liminf_{j\to\infty}\mathbf{M}(S_j)
\le
\limsup_{j\to\infty}\mathbf{M}(S_j)
\le
\mathbf{M}(\llbracket\Sigma\rrbracket).\] Hence \(\mathbf{M}(S_j)\to \mathbf{M}(\llbracket\Sigma\rrbracket).\) Consequently, \(S_j\to\llbracket\Sigma\rrbracket\) in relative flat
topology together with convergence of masses. In particular, the associated mass measures converge weakly: \[\|S_j\|\rightharpoonup \mathcal{H}^n\llcorner\Sigma.\] Since \(\Sigma\subset
V\), it follows that \[\|S_j\|(N\setminus V)\to0.\]
We now construct an exterior replacement. Define \[m_j:=
\inf\left\{
\mathbf{M}(T):
\begin{array}{l}
T\in\mathcal{Z}_n(N,\partial N;\mathbb{Z}_2),\\
{}[T]=[\llbracket\Sigma\rrbracket]\in H_n(N,\partial N;\mathbb{Z}_2),\\
\operatorname{spt}(T-S_j)\subset N\setminus V
\end{array}
\right\}.\] The competitor \(S_j\) itself belongs to this class, and hence \[m_j\le \mathbf{M}(S_j)\le \mathbf{M}(\llbracket\Sigma\rrbracket).\] By compactness together with lower
semicontinuity of mass, there exists a minimizer \[T_j\in\mathcal{Z}_n(N,\partial N;\mathbb{Z}_2)\] such that \[\mathbf{M}(T_j)=m_j,\quad[T_j]=[\llbracket\Sigma\rrbracket]
\quad\text{in }H_n(N,\partial N;\mathbb{Z}_2),\quad\text{and}\quad\operatorname{spt}(T_j-S_j)\subset N\setminus V.\] In particular, \(T_j=S_j\) in \(V\).
We claim first that \(T_j\to \llbracket\Sigma\rrbracket\) in the relative flat topology, and that \(\mathbf{M}(T_j)\to\mathbf{M}(\llbracket\Sigma\rrbracket).\) Indeed, since \(T_j=S_j\) in \(V\), we have \[\|T_j\|(V)=\|S_j\|(V).\] Moreover, \[\mathbf{M}(T_j)\le \mathbf{M}(S_j).\] Therefore \[\|T_j\|(N\setminus V)
=
\mathbf{M}(T_j)-\|T_j\|(V)
\le
\mathbf{M}(S_j)-\|S_j\|(V)
=
\|S_j\|(N\setminus V).\] Since \(\|S_j\|(N\setminus V)\to0,\) we get \(\|T_j\|(N\setminus V)\to0.\) Also, \[\mathbf{M}(S_j-T_j)
\le
\|S_j\|(N\setminus V)+\|T_j\|(N\setminus V)
\to0.\] Hence \[\mathcal{F}_{\mathrm{rel}}^N(T_j-S_j)\le \mathbf{M}(T_j-S_j)\to0.\] Since \(S_j\to\llbracket\Sigma\rrbracket\) in relative flat topology, it follows that \[T_j\to\llbracket\Sigma\rrbracket\qquad\text{in relative flat topology.}\] Finally, from lower semicontinuity and \(\mathbf{M}(T_j)\le \mathbf{M}(S_j)\), we obtain \[\mathbf{M}(\llbracket\Sigma\rrbracket)
\le
\liminf_{j\to\infty}\mathbf{M}(T_j)
\le
\limsup_{j\to\infty}\mathbf{M}(T_j)
\le
\limsup_{j\to\infty}\mathbf{M}(S_j)
=
\mathbf{M}(\llbracket\Sigma\rrbracket).\] Thus \(\mathbf{M}(T_j)\to\mathbf{M}(\llbracket\Sigma\rrbracket).\)
Next we prove that, \[\operatorname{spt}T_j\subset K_r\qquad\text{for all sufficiently large \(j\).}\] Since \(T_j\) minimizes mass among competitors that agree with \(S_j\) in \(V\), it is stationary in the free-boundary sense in \(N\setminus \overline{V}.\) More precisely, let \(X\in\mathfrak
X^\partial(N),\) with \(\operatorname{spt}X\Subset N\setminus\overline{V}.\) Let \(\psi_t\) be its flow. For \(|t|\) small, \(\psi_t\) preserves \(\partial N\), and since \(\psi_t=\operatorname{id}\) near \(\overline{V}\), we have \[\operatorname{spt}\bigl((\psi_t)_\#T_j-S_j\bigr)\subset N\setminus V.\] Moreover, \[[(\psi_t)_\#T_j]=[T_j]=[\llbracket\Sigma\rrbracket]
\quad\text{in }H_n(N,\partial N;\mathbb{Z}_2).\] Therefore \((\psi_t)_\#T_j\) is an admissible competitor in the exterior minimization problem defining \(T_j\). Hence \[\mathbf{M}(T_j)\le \mathbf{M}((\psi_t)_\#T_j)\qquad\text{for all sufficiently small \(|t|\).}\] Dividing by \(t\) and letting \(t\to0^\pm\), we obtain \[\delta |T_j|(X)=0\qquad\text{for every such \(X\). }\] Thus \(|T_j|\) has vanishing first variation in \(N\setminus\overline{V}\) with respect to all
free-boundary admissible vector fields.
By the free-boundary monotonicity formula [[12]][22], there exist constants \[\theta_0>0,\qquad r_1>0,\] depending only on \((N,g)\), \(V\), and \(K_r\), such that the following holds: if \[y\in \operatorname{spt}T_j\cap (N\setminus K_r),\] then, since \[\operatorname{dist}(N\setminus
K_r,\overline{V})>0,\] we may choose \(0<\sigma<r_1\), independent of \(j\), such that \[B_\sigma(y)\cap N\subset N\setminus\overline{V}.\]
If \(y\in N^\circ\), the usual monotonicity formula gives \[\|T_j\|(B_\sigma(y))\ge \theta_0\sigma^n.\] If \(y\in\partial N\), the free-boundary
monotonicity formula gives the analogous estimate \[\|T_j\|(B_\sigma(y)\cap N)\ge \theta_0\sigma^n.\] In either case, there is a constant \(c_0>0\) independent of \(j\) such that \(\|T_j\|(N\setminus V)\ge c_0.\) That contradicts with \(\|T_j\|(N\setminus V)\to0.\) This contradiction shows that \[\operatorname{spt}T_j\cap(N\setminus K_r)=\varnothing\] for all sufficiently large \(j\). Hence \(\operatorname{spt}T_j\subset K_r.\)
We now show that \(T_j\) represents the same local relative homology class as \(\llbracket\Sigma\rrbracket\) in \((K_r,A_r)\). Since \(T_j\) is supported in \(K_r\), we can use the deformation \[H:[0,1]\times K_r\to K_r,
\qquad
H(s,F(x,t))=F(x,(1-s)t),\] which satisfies \[H(0,\cdot)=\operatorname{id}_{K_r},
\qquad
H(1,\cdot)=\pi_r,\qquad\text{and}\qquad H(s,A_r)\subset A_r.\] Therefore \(T_j\) is homologous in \((K_r,A_r)\) to \((\pi_r)_\#T_j.\) Since \((\pi_r)_\#T_j\) is a relative \(n\)-cycle in \((\Sigma,\partial\Sigma)\), the \(\mathbb{Z}_2\) constancy theorem gives, on each
connected component \(\Sigma_\ell\) of \(\Sigma\), \[(\pi_r)_\#T_j
=
\sum_\ell m_{\ell,j}\llbracket\Sigma_\ell\rrbracket,
\qquad
m_{\ell,j}\in\{0,1\}.\] On the other hand, since \(T_j\to\llbracket\Sigma\rrbracket\) in relative flat topology and \(\pi_r\) is Lipschitz, we also have \((\pi_r)_\#T_j\to \llbracket\Sigma\rrbracket\) in relative flat topology. Hence, for all sufficiently large \(j\), \(m_{\ell,j}=1\) for every connected component
\(\Sigma_\ell\). Thus \((\pi_r)_\#T_j=\llbracket\Sigma\rrbracket,\) and consequently \[[T_j]=[\llbracket\Sigma\rrbracket]
\quad\text{in }H_n(K_r,A_r;\mathbb{Z}_2).\]
We can now apply Theorem 4 to \(T_j\), because \[T_j\in \mathcal{Z}_n(K_r,A_r;\mathbb{Z}_2)\qquad\text{and}\qquad
[T_j]=[\llbracket\Sigma\rrbracket]
\quad\text{in }H_n(K_r,A_r;\mathbb{Z}_2).\] Therefore \[\mathbf{M}(T_j)\ge \mathbf{M}(\llbracket\Sigma\rrbracket).\] But by construction, \[\mathbf{M}(T_j)\le \mathbf{M}(S_j)\le
\mathbf{M}(\llbracket\Sigma\rrbracket).\] Hence \[\mathbf{M}(T_j)=\mathbf{M}(\llbracket\Sigma\rrbracket).\] By the equality case in Theorem 4, we
get \(T_j=\llbracket\Sigma\rrbracket\) as a relative cycle in \((K_r,A_r)\), hence also in \((N,\partial N)\).
Since \[\operatorname{spt}(T_j-S_j)\subset N\setminus V\qquad\text{and}\qquad T_j=\llbracket\Sigma\rrbracket\subset V,\] we have \[S_j=\llbracket\Sigma\rrbracket
\quad\text{inside }V.\] Therefore \[\mathbf{M}(S_j)
=
\mathbf{M}(\llbracket\Sigma\rrbracket)
+
\|S_j\|(N\setminus V).\] But \[\mathbf{M}(S_j)\le \mathbf{M}(\llbracket\Sigma\rrbracket).\] Thus \[\|S_j\|(N\setminus V)=0,\] and hence \[S_j=\llbracket\Sigma\rrbracket.\] This contradicts the choice of \(S_j\ne\llbracket\Sigma\rrbracket\). The proof is complete. ◻
4.2 Application to the First Free-boundary Width↩︎
We briefly recall the min–max notation used in the application. We do not need the full construction of the free-boundary min-max theory, and only use its standard consequences. For full construction and related results, see [2], [3], [22].
Let \(\mathcal{Z}_n(N,\partial N;\mathbb{Z}_2)\) denote the space of relative \(n\)-cycles endowed with the flat, or \(\mathbf{F}\)–topology. A
one-sweepout is a continuous map \(\Phi:X\to \mathcal{Z}_n(N,\partial N;\mathbb{Z}_2)\) from a finite cubical complex \(X\) such that \[\Phi^*(\lambda)\neq 0
\quad\text{in }H^1(X;\mathbb{Z}_2),\qquad\text{where }\lambda\in H^1(\mathcal{Z}_n(N,\partial N;\mathbb{Z}_2);\mathbb{Z}_2)\] denotes the fundamental cohomology class of the relative cycle space given by the Almgren isomorphism [25]. The first free-boundary width is defined by \[\omega_1(N,g)
:=
\inf_{\Phi}
\sup_{x\in X}\mathbf{M}(\Phi(x)),\] where the infimum is taken over all one-sweepouts.
A minimizing sequence for \(\omega_1(N,g)\) is a sequence of one-sweepouts \[\Phi_j:X_j\to\mathcal{Z}_n(N,\partial N;\mathbb{Z}_2)\] such that \[\lim_{j\to\infty}\sup_{x\in X_j}\mathbf{M}(\Phi_j(x))
=
\omega_1(N,g).\] After applying the pull-tight procedure, the associated critical set consists of stationary integral varifolds obtained as varifold limits of slices \(|\Phi_j(x_j)|\) with \[\mathbf{M}(\Phi_j(x_j))\to \omega_1(N,g).\]
We shall use the following standard consequences of the free-boundary min–max theory. By the strong bumpy metric theorem of [2], for a generic metric, every
embedded free-boundary minimal hypersurface is proper and non-degenerate. Thus, in the generic setting considered below, embedded free-boundary minimal hypersurfaces produced by the min–max theory are automatically properly embedded. Moreover, by the
multiplicity-one theorem in same article, for a generic metric in dimensions \(3\le n+1\le 7\), there exists a smooth, properly embedded, multiplicity-one free-boundary minimal hypersurface \(\Gamma\) realizing \(\omega_1(N,g)\). By the Morse index upper bound of [3], such a
multiplicity-one critical hypersurface satisfies \[\operatorname{index}(\Gamma)\le 1.\] If \(\Gamma\) is disconnected, say \[\Gamma=\Gamma_1\cup\cdots\cup\Gamma_q,\] we use the convention \[\operatorname{index}(\Gamma)
:=
\sum_{\ell=1}^q \operatorname{index}(\Gamma_\ell).\]
We now explain why the flat-neighborhood version is useful for the Morse index problem in free-boundary min–max theory. In the first-width construction, the multiplicity-one theorem of Sun–Wang–Zhou provides a multiplicity-one free-boundary min–max
critical cycle realizing \(\omega_1(N,g)\)[2]. The general index upper bound of Guang–Li–Wang–Zhou gives total index at most
one [3]. Thus the only remaining issue is to rule out the possibility that this multiplicity-one critical cycle is stable.
This is precisely where Theorem 15 enters. If such a critical cycle were stable, then, under the bumpy metric
assumption, it would be strictly stable. Theorem 15 would then give a flat neighborhood which is avoidable by all
sufficiently good nontrivial 1-sweepouts. This contradicts the fact that the cycle occurs as a min–max critical element. Hence the critical cycle must have positive index, and the index upper bound forces its total index to be one.
Theorem 16 (Index one for a multiplicity-one realization of \(\omega_1\)). Let \(3\le n+1\le 7\), and let \((N^{n+1},g)\) be a compact
Riemannian manifold with boundary. Assume that the bumpy metric \(g\) is chosen generically so that the multiplicity-one theorem of Sun–Wang–Zhou [2] holds.
Let \(L:=\omega_1(N,g).\) Then there exists a smooth, properly embedded, multiplicity-one, 2-sided free-boundary minimal hypersurface \[\Gamma=\Gamma_1+\cdots+\Gamma_q\] such that \[\mathbf{M}(\Gamma)=L\qquad\text{and}\qquad\sum_{\ell=1}^q \operatorname{index}(\Gamma_\ell)=1.\] In particular, if this hypersurface is connected, then \[\operatorname{index}(\Gamma)=1.\]
Proof. By the free-boundary multiplicity-one theorem and the strong bumpy metric theorem of Sun–Wang–Zhou [2], there exists a pulled-tight minimizing
sequence \[\Phi_j:X\to\mathcal{Z}_n(N,\partial N;\mathbb{Z}_2)\] for the first width \(L=\omega_1(N,g),\) and there exists a smooth, properly embedded, multiplicity-one, 2-sided
free-boundary minimal hypersurface \[\Gamma=\Gamma_1+\cdots+\Gamma_q\] in the critical set of this sequence such that \(\mathbf{M}(\Gamma)=L.\) Here \(X\)
is connected and represents a nontrivial one-sweepout class.
By the free-boundary Morse index upper bound of Guang–Li–Wang–Zhou [3], the total Morse index of \(\Gamma\) satisfies \[\operatorname{index}(\Gamma)
:=
\sum_{\ell=1}^q \operatorname{index}(\Gamma_\ell)
\le 1.\] It remains to prove that \[\operatorname{index}(\Gamma)\ne0.\]
Suppose, by contradiction, that \(\operatorname{index}(\Gamma)=0.\) Then every component \(\Gamma_\ell\) is stable. Since \(g\) is free-boundary bumpy, no
component admits a nontrivial free-boundary Jacobi field. Hence every component is strictly stable, and therefore the whole multiplicity-one cycle \(\Gamma\) is strictly stable.
By Theorem 15, there exists a relative flat neighborhood \(\mathcal{U}\) of \(\llbracket\Gamma\rrbracket\) such that \(\llbracket\Gamma\rrbracket\) is the unique mass minimizer in its local relative homology class inside \(\mathcal{U}\).
Shrinking \(\mathcal{U}\) if necessary, we may choose another relative flat neighborhood \[\mathcal{U}'\Subset\mathcal{U}\qquad\text{and}\qquad\delta>0\] such that \[T\in \mathcal{U}\setminus \mathcal{U}'
\quad\Longrightarrow\quad
\mathbf{M}(T)\ge L+2\delta.\] Indeed, like in Corollary 2, this is an immidiate consequence of the local minimizing property.
Since \(\{\Phi_j\}\) is minimizing for \(\omega_1(N,g)\), for all sufficiently large \(j\) we have \[\sup_{x\in
X}\mathbf{M}(\Phi_j(x))<L+\delta.\]
We claim that, for all sufficiently large \(j\), \[\Phi_j(X)\cap\mathcal{U}'=\varnothing.\] Suppose not. Then there exist \(j\) and \(x_0\in X\) such that \(\Phi_j(x_0)\in\mathcal{U}'.\) Since \(X\) is connected, either \[\Phi_j(X)\subset\mathcal{U},\] or
else \(\Phi_j(X)\) intersects \[\mathcal{U}\setminus\mathcal{U}'.\] In the second case, there exists \(x_1\in X\) such that \[\Phi_j(x_1)\in\mathcal{U}\setminus\mathcal{U}'.\] Hence \[\mathbf{M}(\Phi_j(x_1))\ge L+2\delta,\] contradicting \[\sup_{x\in
X}\mathbf{M}(\Phi_j(x))<L+\delta.\] Thus the only remaining possibility is \(\Phi_j(X)\subset\mathcal{U}.\)
But \(\mathcal{U}\) is a sufficiently small relative flat neighborhood. Hence it is trivial for the first sweepout class: equivalently, the Almgren cohomology class detecting one-sweepouts vanishes on \(\mathcal{U}\). Therefore a map whose image is contained in \(\mathcal{U}\) cannot represent a nontrivial one-sweepout. This contradicts the fact that \(\Phi_j\)
belongs to the minimizing sequence for \(\omega_1(N,g)\). The claim follows.
On the other hand, since \(\Gamma\) belongs to the critical set of \(\{\Phi_j\}\), there exist \[j_k\to\infty,
\qquad
x_k\in X,\] such that \[\mathbf{M}(\Phi_{j_k}(x_k))\to L\qquad\text{and}\qquad |\Phi_{j_k}(x_k)|\to |\Gamma|\] as varifolds. Since \(\Gamma\) has multiplicity one, and since the
masses converge to \(\mathbf{M}(\Gamma)\), the associated relative cycles converge to \(\llbracket\Gamma\rrbracket\) in the relative flat topology, after passing to a subsequence.
Equivalently, \[\Phi_{j_k}(x_k)\to \llbracket\Gamma\rrbracket
\qquad
\text{in } \mathcal{F}_{\mathrm{rel}}.\] Thus, for \(k\) sufficiently large, \[\Phi_{j_k}(x_k)\in\mathcal{U}',\] contradicting the claim that \(\Phi_j(X)\cap\mathcal{U}'=\varnothing\) for all large \(j\).
Therefore \(\operatorname{index}(\Gamma)\ne0.\) Combining this with \(\operatorname{index}(\Gamma)\le1,\) we obtain \[\operatorname{index}(\Gamma)=1.\]
That is, \[\sum_{\ell=1}^q \operatorname{index}(\Gamma_\ell)=1.\] The theorem follows. ◻
The preceding application should be regarded as a first use of the relative flat-neighborhood local minimality theorem. We expect that the theorem may be useful in other geometric variational problems where one needs to pass from stability to local
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