June 14, 2026
We classify any entire area-minimizing surface \(M^2\subset\mathbb{R}^n\) with density \(2\) at infinity as either planar or algebraic, namely the zero set of a quadratic holomorphic polynomial (up to rigid motion) and contained within an affine copy of \(\mathbb{R}^4\). In particular, \(M\) is either planar, or smooth, multiplicity one, and genus zero.
Motivated by the interactions between area-minimizing surfaces and holomorphic subvarieties, we are interested in the following question:
Question 1. What is the structure of an entire, area-minimizing surface in \(\mathbb{R}^n\)?
It is well-known that any holomorphic subvariety of \(\mathbb{C}^n\) is area-minimizing in its homology class. To what extent the converse holds, namely whether an entire area-minimizing surface in \(\mathbb{R}^n\) is necessarily holomorphic (up to rigid motion) has been asked by numerous authors [1]–[4].
Globally-defined minimal surfaces often exhibit strong rigidity properties, and perhaps the most famous example of this is the Bernstein problem for entire, codimension-one minimal graphs (or, more generally, for area-minimizing or stable hypersurfaces) which asserts that such hypersurfaces must be flat in low dimensions [5]–[16]. Aside from being a natural classification question, understanding globally-defined minimal surfaces is in some sense “dual” to understanding the local degeneration of solutions, as one expects the small-scale picture of a surface near the formation of a singularity to be modeled on a global solution.
A starting point for Question 1 is the work of Morgan [17], which classified dilation-invariant (“conical”) area-minimizers \(T^2\subset\mathbb{R}^n\) as unions of holomorphic planes (holomorphic relative to some fixed orthogonal complex structure). In the special case \(n=4\), i.e. \(M^2\subset\mathbb{R}^4\equiv \mathbb{C}^2\), Micallef [1], [18] gave a remarkable answer to Question 1 by showing that in fact if \(M\) is a complete, connected, oriented, parabolic, stable (branched) minimal surface, then it is holomorphic (up to a real rigid motion). In [19] we made explicit the further observation that if such \(M^2\subset\mathbb{R}^4\) additionally has quadratic area growth, then in fact \(M\) must be algebraic.
In higher codimension (\(n \geq 5\)), most progress has only been made when the surface is known a priori to have genus \(0\) or \(1\), primarily because of nice splitting properties of the complexified normal and tangent bundles. Micallef [1] showed that if \(M^2\subset\mathbb{R}^n\) is complete, minimal, stable, genus \(0\), and has finite total curvature, then \(M\) must again be holomorphic up to a rigid motion, and is secretly contained within some copy of \(\mathbb{C}^k\equiv \mathbb{R}^{2k}\). Fraser–Schoen [20] proved a holomorphicity result analogous to Micallef for genus \(1\) surfaces which are “covering stable,” and recently Sagman–Thomas-Rene [21] extended the results of both Micallef and Fraser–Schoen by removing the assumptions of completeness and finite total curvature. To illustrate the subtlety in general codimension, even in the case where the surface is smooth, there are examples of genus \(2\) stable minimal immersions into \(\mathbb{R}^{22}\) with finite total curvature which are not holomorphic with respect to any orthogonal complex structure (see [22], [23]).
For the general situation of area-minimizing surfaces as in Question 1 (precisely, “mass-minimizing integral \(2\)-currents \(T\) without boundary”), it is known by the work of Rivière [24] (see Theorem 6) that if \(T\) has quadratic area growth, then \(T\) must be asymptotic to a union of planes, possibly with multiplicity (and these planes must be jointly holomorphic for some orthogonal complex structure by [17]). The total number of planes, counted with multiplicity, is equal to the density \(\Theta_T(\infty) \in \mathbb{N}\) of \(T\) at infinity. The quantity \(\Theta_T(\infty)\) may be viewed as a measure of the complexity of \(T\). The case \(\Theta_T(\infty)=1\) is trivial, as then \(T\) is necessarily a multiplicity-one plane (seen by, e.g., the monotonicity formula). For density \(\Theta_T(\infty)=2\), there are non-trivial examples given by the zero set of any quadratic holomorphic polynomial in \(\mathbb{C}^2\), e.g. \[\{(z,w)\in \mathbb{C}^2: z^2=w\} \quad \text{or}\quad \{(z,w)\in \mathbb{C}^2:zw=1\}.\]
Thus, area-minimizing surfaces \(T\subset\mathbb{R}^n\) with \(\Theta_T(\infty)=2\) are in some sense the “simplest” non-trivial area-minimizing surfaces. Even so, their study (in particular in regards Question 1), can be quite tricky, as a priori such surfaces could be very structurally complicated. A key difficulty which illustrates the non-trivial subtleties of the problem is seen in the instance where \(T\) is asymptotic to a multiplicity \(2\) plane, as one could a priori imagine a situation where \(T\) looks like two parallel planes connected by an infinite sequence of “necks” or “handles” which drift off to infinity. This problem of how to deal with “infinite topology” is a common difficulty in the study of minimal surfaces.
Our main result is a resolution of Question 1 when \(\Theta_T(\infty)=2\). We completely classify such \(T\), showing that \(T\) must either be planar or must coincide with the zero set of a degree \(2\) holomorphic polynomial in some copy of \(\mathbb{C}^2\).
Theorem 1. Let \(T\) be an area-minimizing integral \(2\)-current in \(\mathbb{R}^{2+m}\) (for \(m\geq 2\)) satisfying \(\partial T = 0\) and \(\Theta_T(\infty)=2\). Identify \(\mathbb{C}^2\equiv \mathbb{R}^4\times\{0^{m-2}\}\subseteq\mathbb{R}^{2+m}\). Then \(T\) takes one of the following forms:
\(T = 2\llbracket P\rrbracket\) for some affine \(2\)-plane \(P\subset\mathbb{R}^{2+m}\);
\(T = A_\#\llbracket p_1+P_1\rrbracket + A_\#\llbracket p_2+P_2\rrbracket\) for \(P_1,P_2\) complex \(2\)-planes in \(\mathbb{C}^2\), \(p_1,p_2\in \mathbb{R}^{2+m}\), and \(A\in O(2+m)\);
\(T = A_\#\llbracket p + \{(z,w)\in\mathbb{C}^2: az^2+bzw+cw^2+dz+ew+f = 0\}\rrbracket\) for some \(a,b,c,d,e,f\in \mathbb{C}\), \(p\in \mathbb{R}^{2+m}\), and \(A\in O(2+m)\).
In particular, \(T\) is either planar, or \(T\) is smooth, multiplicity one, genus zero, and is (up to a rigid motion) the zero set of a quadratic holomorphic polynomial.
We emphasize that Theorem 1 proves that \(T\) has genus \(0\) using the weak asymptotics at infinity. Theorem 1 also shows that density \(2\) at infinity is strictly stronger than having genus \(0\), as one can easily construct examples of genus \(0\) holomorphic curves in \(\mathbb{C}^n\) which are not contained within a copy of \(\mathbb{C}^2\), such as the image of \(z\mapsto (z,z^2,z^3)\in \mathbb{C}^3\).
It is tempting to conjecture that any area-minimizing surface in \(\mathbb{R}^n\) with quadratic area growth must be algebraic, as is the case in \(\mathbb{R}^4\) (see [19]). However, the counterexamples in [22] of stable minimal surfaces in high codimension show that the problem is quite subtle, and would strongly use the area-minimizing property.
Nonetheless, our methods are not purely restricted to the case \(\Theta_T(\infty)=2\). In fact, for general \(\Theta_T(\infty) = Q\in \mathbb{N}\), we are able to classify the “simplest” \(T\) which are asymptotic to a single plane \(P\) with multiplicity \(Q\). Indeed, in this situation we are able to show that it is possible to adapt the planar frequency function of Krummel–Wickramasekera to analyze the behavior of the current, but now at infinity. This allows us to associate another quantity which measures the “complexity” of \(T\), namely the planar frequency value \(N_{T,P}(\infty)\) of \(T\) at infinity, which we show takes values \(N_{T,P}(\infty)\in \{p/q: p\in \{1,\dotsc,q-1\}, q\in\{2,\dotsc,Q\}\}\). In the “simplest” case \(N_{T,P}(\infty)=1/Q\) (which, when \(Q = 2\), is the only possible option), we are able to show that \(T\) must be contained within some affine copy of \(\mathbb{R}^4\equiv \mathbb{C}^2\), which implies from [19] that \(T\) must be algebraic. In particular, up to a rigid motion and dilation, such \(T\) are given by the zero set of the polynomial \[\{(z,w)\in \mathbb{C}^2: z^Q=w\},\] and so \(T\) must be embedded with genus \(0\). Unlike when \(Q=2\), in the case \(Q\geq 3\) a priori \(T\) could have interior singularities, including branch points, which we a posteriori show do not exist.
Outline of the Proof: Let us describe our strategy to proving Theorem 1. The overall goal is to prove that such \(T\) must be supported in some affine copy of \(\mathbb{R}^4\), as then the problem is reduced to the classification problem in \(\mathbb{R}^4\) as considered by [1], [19].
The initial observation is that from [24] we know that \(T\) scale-invariantly decays to some unique tangent cone at infinity, which is necessarily a sum of two, possibly identical, planes. Therefore, there are two cases to consider, depending on whether \(T\) is asymptotic to two distinct (necessarily transverse) planes \(P_1,P_2\), or \(T\) is asymptotic to a single plane \(P\) with multiplicity \(2\).
The first key step is to prove \(T\) has good asymptotic behavior (is “regular at infinity”). The difficult case for this is when \(T\) is asymptotic to the multiplicity-2 plane \(2\llbracket P\rrbracket\). There is not, in general, an Allard-type regularity theorem for area-minimizing surfaces of codimension \(\geq 2\) near a plane with multiplicity (and indeed as previously mentioned a key difficulty is that a priori such \(T\) might exhibit infinite topology drifting off to infinity). Nonetheless, we are able to prove a multiplicity-\(2\) (and even multiplicity-3) sheeting theorem does hold in this setting (Lemma 9)
The primary tool for our sheeting theorem is an adaptation of the intrinsic planar frequency function \(N_{T,P}(\rho)\) of [25], which is defined for any \(T\), but resembles the standard frequency function \(N_v(\rho)\) when \(T\) is well-approximated by a Dirichlet-minimizer \(v\). We show that if both the area density \(\Theta_T(\rho)\) and the planar frequency \(N_{T,P}(\rho)\) are suitably pinched, then, outside of some ball, \(T\) splits into a smooth, multiplicity-one, \(2\)-valued graph modeled on a \(1/2\)-homogeneous Dirichlet-minimizer. We state this below (see Sections 1 & 2 for details on notation).
For \(T\) close to a plane of arbitrary multiplicity, the analogue of Lemma 1 implies a separation-type theorem – see Remark 1.
To apply Lemma 1, we generalize the frequency monotonicity formula of [25] (originally used for \(T\) decaying to \(P\) near \(0\)) to show that the planar frequency \(N_{T,P}(\rho)\) is monotone almost-increasing near infinity (Theorem 7). In fact, we prove a general scale-invariant differential inequality for \(N_{T,P}(\rho)\) in terms of the \(L^2\) height excess \(E_{T,P}(0,\rho)\) of \(T\) relative to \(P\), which implies almost-monotonicity in a variety of potential cases. The upshot is that for an area-minimizing \(T\) asymptotic to \(2\llbracket P\rrbracket\), we can show that, for large scales, \(T\) looks like the graph over \(P\) of a function like \[\label{eqn:intro-1} ar^{1/2}\cos(\theta/2)+br^{1/2}\sin(\theta/2) + c\log(r) + d + O(r^{-1/2}),\tag{1}\] for \(a,b,c,d\in P^\perp\) and \(a,b\) linearly independent. Here, \((r,\theta)\) are polar coordinates in the \(2\)-plane \(P\). We stress that almost-monotonicity of planar frequency near infinity is the mechanism which rules out pathological behavior of \(T\) at large scales, as it allows one to show that the asymptotic behavior of \(T\) is modeled on its tangent maps at infinity (which, at this stage of the proof, are not known to be unique), which are \(2\)-valued Dirichlet-minimizers that are non-zero, average-free, and homogeneous (of degree \(<1\)).
In the easier case when \(T\) is asymptotic to distinct planes \(\llbracket P_1\rrbracket + \llbracket P_2\rrbracket\), then from Allard’s regularity theorem we can deduce that each end of \(T\) looks like the graph over \(P_i\) of a function like \[\label{eqn:intro-2} c_i\log(r)+d_i+O(r^{-1}),\tag{2}\] for \(c_i,d_i\in P_i^\perp\) and \(r\) the radial variable in \(P_i\). So, in either case, we have precise asymptotics, and one might refer to such \(T\) as being “regular at infinity”.
The second key ingredient is a Liouville-type theorem which shows \(T\) that lies in some affine copy of \(\mathbb{R}^4\) (Theorem 9). In the first case 1 , the candidate \(4\)-space is \(W:= P\oplus \mathbb{R}a \oplus \mathbb{R}b\), whilst in the second case 2 the candidate \(4\)-space is \(W:=P_1\oplus P_2\). We use the area-minimizing property to show that the \(\log\) terms in 1 or 2 must lie in \(W\) (i.e. \(c\in W\) or \(c_1,c_2\in W\)). This is done by using a competitor argument, as one can build a competitor with less area if this were not the case. From the expansions 1 , 2 , we then get that the coordinate functions in the \(W^\perp\) are bounded, and using harmonicity of the coordinate functions and parabolicity of \(T\) it follows that the \(W^\perp\) directions must all be constant in \(T\), and hence \(T\) is supported in some translate of \(W\).
We highlight that the principle of bounding codimension in terms of density is very general, and applies to general densities at infinity as long as \(T\) is indecomposable and regular at infinity. We state this below.
It’s worth comparing and contrasting Theorem 2 to the codimension bound of [26], which implies that any (smooth) stationary \(T^n \subset \mathbb{R}^{n+m}\) is contained in an affine subspace of dimension \(\leq c(n) \Theta_T(\infty)\),1 for some dimensional constant \(c(n)\). One can interpret our Theorem 2 as saying that when \(n = 2\) and \(T\) is area-minimizing (and has reasonable asymptotics), then the sharp constant \(c(n)\) is \(2\) (though we also remark our proof strategy is very different from [26]). Having the sharp bound \(2\Theta_T(\infty)\) is obviously crucial for our argument, and obtaining this bound also requires heavily the area-minimizing property. Indeed, the following example shows that the bound \(2\Theta_T(\infty)\) can fail for complete, embedded, minimal \(2\)-surfaces which are merely stationary.
Example 1. If \(F : \mathbb{C}\setminus \{0\} \to \mathbb{R}^5\) is the map \[F(z) = \mathrm{Re} \left( -(z^{-1} + z/4), -\sqrt{-1}(z^{-1} - z/4), z, \sqrt{-1}z, \log(z) \right),\] then by [27] (see also [28]) \(F\) defines a complete, embedded, minimal annulus \(M^2 \subset \mathbb{R}^5\) (with total curvature \(-4\pi\)) which is not contained in any affine \(\mathbb{R}^4\). It’s easy to see that each end has multiplicity-one, and so \(\Theta_M(\infty) = 2\).
We expect Theorem 2 to hold for any indecomposable area-minimizing integral \(2\)-current \(T\) with \(\Theta_T(\infty)<\infty\). It is worth noticing that Theorem 2 holds (and is sharp) when \(T\) is an irreducible algebraic curve in \(\mathbb{C}^n\), as in this case each independent projective complex coordinate contributes one complex dimension to the canonical line bundle, but on the other hand this line bundle has complex dimension \(\leq \Theta_T(\infty) + 1\).
More generally, we would expect that any area-minimizing \(2\)-current \(T\) with finite density to be regular at infinity, by analogy with Chang’s regularity theorem [29], [30] which says that near any (interior) singular point \(T\) is a branched minimal immersion. If the tangent cone of \(T\) at infinity is multiplicity-one (by Allard) or is a density \(\leq 3\) plane (by Lemma 9, Remark 1), then \(T\) is regular at infinity. Likely there is a notion of a (branched) center manifold at infinity, however one initial issue is that, unlike the local picture, it is not clear how or whether one can reduce to studying \(T\) near a single plane with multiplicity (as illustrated by \(\{zw=1\}\subset\mathbb{C}^2\)). Regularity at infinity would immediately imply finite total curvature and finite genus, and would be amenable to proving a priori bounds on the total curvature also.
Acknowledgments: We thank Gábor Székelyhidi, Brian White, and Neshan Wickramasekera for inspiring conversations. N.E. was support by NSF grant DMS-2506700 and a Simons Foundation travel award. This research was conducted during the period P.M. served as a Clay Research Fellow.
We work with \(n\)-currents and \(n\)-varifolds in \(\mathbb{R}^{n+m}\), with our main results concerning the case \(n=2\) and several auxiliary results holding for general \(n\). We make use of standard notation and concepts from geometric measure theory, referring the reader to [31] for general background, and [32]–[34] for background on multi-valued functions and the theory of high-codimension area-minimizing currents. We will write \(c\) for a generic constant \(c\geq 1\), and \(\epsilon,\delta,\eta\) for generic (positive) constants \(<1\). Named constants, such as \(c_1,c_2,\dotsc\), are fixed and do not change in value.
We identify \(\mathbb{R}^n\cong \mathbb{R}^n\times\{0^m\}\subset \mathbb{R}^{n+m}\). Given \(x\in \mathbb{R}^{n+m}\), the standard open Euclidean ball of radius \(\rho>0\) centered at \(x\) is \(B_\rho(x):= \{y\in \mathbb{R}^{n+m}:|y-x|<\rho\}\), and when \(x\in \mathbb{R}^n\) we write \(B_\rho^n(x):= B_\rho(x)\cap \mathbb{R}^n\) for the corresponding \(n\)-dimensional ball in \(\mathbb{R}^n\). For \(0<\sigma<\rho\) write \(A_{\rho,\sigma}(x):= B_\rho(x)\setminus \overline{B_\sigma(x)}\) for the open annulus with inner radius \(\sigma\) and outer radius \(\rho\), and \(A^n_{\rho,\sigma}(x):= A_{\rho,\sigma}(x)\cap \mathbb{R}^n\). Write \(B_\rho := B_\rho(0)\), and define analogously \(B_\rho^n\), \(A_{\rho,\sigma}\), and \(A^n_{\rho,\sigma}\). We will always write \(d_H(A,B)\) for the Hausdorff distance between subsets \(A,B\subset\mathbb{R}^{n+m}\), and \(d_A(x)\) for the distance between \(x\in \mathbb{R}^{n+m}\) and a subset \(A\subset\mathbb{R}^{n+m}\).
Let \(P\) be an oriented \(n\)-plane (typically we take \(P = \mathbb{R}^m\times\{0^m\}\)). Write \(\pi_P\) for the orthogonal projection to \(P\), and \(\pi_P^\perp\) for the orthogonal projection to \(P^\perp\). Given \(z\in \mathbb{R}^{n+m}\), we write \(d_{z+P}(x):= |\pi_P^\perp(x-z)|\) for the distance of \(x\) to the affine plane \(z+P\). We also define the open cylinder over \(P\) of radius \(\rho\) and center \(z\) by \[C_\rho(z,P):= \{x\in \mathbb{R}^{n+m}:|\pi_P(x-z)|<\rho\}.\] For shorthand, we write \(C_\rho(z) := C_\rho(z,\mathbb{R}^n\times\{0^m\})\) and \(C_\rho := C_\rho(0)\).
We will often consider the special case of a \(Q\)-valued function which lifts to a single-valued function on a \(Q\)-branched copy of \(\mathbb{R}^2\setminus\{0\}\). As such, we introduce specialized notation for this setting. Let us define \(QS^1:= [0,2\pi Q]/\sim\), where \(\sim\) is the equivalence relation on \([0,2\pi Q]\) with the only extra relation being \(0\sim 2\pi Q\). We then write \(v:Q\mathbb{R}^2\to \mathbb{R}^m\) to mean a function \(v:(0,\infty)\times QS^1\to \mathbb{R}^m\) (note that \(v\) is not defined at \(0\in \mathbb{R}^2\)), and similarly \(v:Q\mathbb{R}^2\cap A_{b,a}\to \mathbb{R}^m\) for a function \(v:(a,b)\times QS^1\to \mathbb{R}^m\).
One can interpret \(v:Q\mathbb{R}^2\to \mathbb{R}^m\) as a function defined on a \(Q\)-branched cover of \(\mathbb{R}^2\setminus\{0\}\), and in particular in a neighborhood of any \((r,\theta)\) with \(r\neq 0\), we can think of \(v\) as being defined on an open subset of \(\mathbb{R}^2\). Away from the origin, we can therefore make sense of the standard Euclidean derivatives \(D^k v\), and talk about \(v\) solving a PDE of the form \(G(D^2v,Dv,v)=0\) by asking \(v\) to solve the PDE locally, as a single-valued function, on each branch.
Given \(v : Q\mathbb{R}^2 \to \mathbb{R}^m\), write \[\label{eqn:prelim-1} \mathrm{graph}_{Q\mathbb{R}^2}(v):= \{(r\cos(\theta),r\sin(\theta),v(r,\theta): r\in (0,\infty)),\, \theta\in QS^1\},\tag{3}\] and similarly define \(\mathrm{graph}_{Q\mathbb{R}^2}(v)\cap A_{b,a}\) for \(v : Q\mathbb{R}^2 \cap A_{b, a} \to \mathbb{R}^m\), except now the radial variable \(r\) obeys \(r\in (a,b)\). We always endow \(\mathrm{graph}_{Q\mathbb{R}^2}(v)\) with the orientation induced by the graphing map 3 . Write \(Jv := \sqrt{ \det(\delta_{ij} + \langle D_i v, D_j v\rangle)} : Q\mathbb{R}^2 \to \mathbb{R}\) for the Jacobian of the graphing map, and for shorthand write \[\int_{Q\mathbb{R}^2 \cap A_{b, a}} v \, dx := \int_{QS^1} \int_a^b v(r, \theta) \, r dr d\theta .\]
We will be particularly interested in \(v:Q\mathbb{R}^2\to \mathbb{R}^m\) which solves the minimal surface system \[\label{eqn:mss} g^{ij}D_{ij}^2v = 0,\tag{4}\] where \(g^{ij} = (g^{-1})_{ij}\) for \(g = (g_{ij})_{ij}\) with \(g_{ij} = \delta_{ij} + \langle D_iv,D_jv\rangle\). This can be recast as a perturbation of the Laplacian, namely \[\label{eqn:mss-rough} \Delta v = F(Dv,D^2v) = Dv\star Dv\star D^2v\tag{5}\] where we use the shorthand \[p \star p \star q \equiv \sum_{\substack{i,j,k,l=1,2 \\\alpha, \beta, \gamma = 1, \ldots, m}} c_{\alpha \beta \gamma}^{ijkl}(p, q) p_i^\alpha p_j^\beta q_{kl}^\gamma,\] for analytic functions \(c^{ijkl}_{\alpha\beta\gamma}(p, q)\)
Given \(f : (a, \infty) \times QS^1 \to \mathbb{R}^m\) (or simply \(f: (a,\infty) \to \mathbb{R}^m\)), we write \(f = O(r^\alpha)\) to mean \(|f(r,\cdot)|\leq Cr^\alpha\) for \(r\gg 1\) and some constant \(C\). Similarly, we write \(f = O_k(r^\alpha)\) to mean \(r^i D^i f = O(r^\alpha)\) for each \(i\in\{0,1,\dotsc,k\}\), and \(f = O_{k,\gamma}(r^\alpha)\) to mean \(f = O_k(r^\alpha)\) and \(r^{k+\gamma}[D^k f]_{\gamma;A_{2r,r}} = O(r^\alpha)\). Finally, write \(f = O_{k,1-}(r^\alpha)\) if \(f = O_{k,\gamma}(r^\alpha)\) for all \(\gamma<1\), and \(f = O(r^{\alpha+})\) if \(f = O(r^{\alpha+\epsilon})\) for all \(\epsilon>0\). Similarly, we define \(O_k(r^{\alpha+})\), \(O_{k,\gamma}(r^{\alpha+})\), and \(O_{k,1-}(r^{\alpha+})\) analogously.
If \(V\) is a stationary integral \(n\)-varifold in \(\mathbb{R}^{n+m}\), the density ratio \(\Theta_V(x, r)\) on the ball \(B_r(x)\) is defined to be \[\Theta_V(x, r) = \frac{\|V\|(B_r(x))}{\omega_n r^n},\] where \(\omega_n\) is the \(n\)-volume of the \(n\)-dimensional unit ball. The classical monotonicity formula says that \(\Theta_V(x, r)\) is increasing in \(r\), and is constant in \(r\) if and only if \(V\) is dilation-invariant centered at \(x\). In particular, the quantities \[\Theta_V(x) := \lim_{r \to 0} \Theta_V(x, r), \quad \Theta_V(\infty) := \lim_{r \to \infty} \Theta_V(0, r)\] always exist (but the latter may be infinite). Integrality implies \(\Theta_V(x) \geq 1\) iff \(x \in \mathrm{spt}\|V\|\).
Likewise, if \(T\) is an area-minimizing integral \(n\)-current with \(\partial T = 0\), then the underlying integral \(n\)-varifold \(|T|\) is stationary, and so one can define \(\Theta_T := \Theta_{|T|}\). A basic fact is that any dilation-invariant area-minimizing \(2\)-current in \(\mathbb{R}^{2+m}\) is supported on a union of transverse planes, and therefore if \(n = 2\) it holds that \(\Theta_T(x), \Theta_T(\infty) \in \mathbb{N}\cup \{ \infty \}\).
We will make use of the following “excess” quantities. Define \[E_{T, P}(z, \rho) := \rho^{-n-2} \int_{C_{\rho}(z, P)} d_{z + P}(x)^2\, d\|T\|(x)\] for the \(L^2\) height excess in the cylinder \(C_\rho(z,P)\), and \[\begin{align} \mathcal{E}_{T, P}(z, \rho) & := \rho^{-n} \|T\|(C_{\rho}(z, P)) - \rho^{-n} \|\pi_{P \#} (T \mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_{\rho}(z, P))\|(B^n_\rho(z)) \\ &\equiv \frac{1}{2\rho^n} \int_{C_\rho(z, P)} |\vec{T} - \vec{P}|^2\, d\|T\| \end{align}\] for the mass excess in the cylinder (with the second alternative form being true up to flipping the sign of the orientation on \(P\), cf. [35]). Here \(\vec{T}(x)\) is the orienting \(n\)-vector for the approximating tangent space \(T_x T\) of \(T\) at \(x\), \(\vec{P}\) the orienting \(n\)-vector for \(P\), and \(|\vec{T} - \vec{P}|\) is the norm induced on the exterior product.
Though we will not name them explicitly, we will also commonly use the \(L^2\) height and tilt excesses in a ball \(B_\rho(y)\) (relative to an affine plane \(z+P\)), defined respectively as \[\rho^{-n-2} \int_{B_\rho(y)} d_{z + P}(x)^2\, d\|T\|(x), \quad \rho^{-n} \int_{B_\rho(y)} \|\pi_T - \pi_P\|^2\, d\|T\|(x).\] Here \(\pi_T|_x\) is the projection onto the approximate tangent space \(T_x T\), and \(\|\pi_T - \pi_P\|\) is the Frobenius norm. The mass excess and tilt excess are both kinds of geometric \(W^{1,2}\)-norms, but the former “sees” orientation while the latter does not. We record the following elementary geometric inequality \[\label{eqn:tilt-by-tilt} \|\pi_T - \pi_P\| \leq c(n, m) |\vec{T} - \vec{P}|.\tag{6}\]
We now record well-known relationships concerning the respective excesses.
Lemma 2 (\(L^2\) controls \(L^\infty\)). Let \(V\) be a stationary integral \(n\)-varifold in \(B_1\). Then for any \(\gamma \in (0, 1)\) we have \[\label{eqn:Linfty-by-L2-concl1} \sup_{x \in \mathrm{spt}\|V\| \cap B_\gamma} d_P(x)^2 \leq c(n, m, \gamma) \int_{B_1} d_P^2\, d\|V\|,\tag{7}\] If instead \(V\) is a stationary integral \(n\)-varifold in \(C_1\), we have \[\label{eqn:Linfty-by-L2-concl2} \sup_{x \in \mathrm{spt}\|V\| \cap C_\gamma} d_P(x)^2 \leq c(n, m, \gamma) \int_{C_1} d_P^2\, d\|V\|,\tag{8}\] and in particular, we have \[\mathrm{spt}\|V\| \cap C_\gamma \subset B_{\gamma+c(n, m,\gamma)E_{V, P}(0, 1)^{1/2}}.\]
Proof. The first inequality is well-known, see e.g. [36]. To see the second inequality, we break into two cases. If \(x \in \mathrm{spt}\|V\| \cap C_\gamma\) satisfies \(d_P(x) \leq \eta \leq 1-\gamma\) for suitably small \(\eta = \eta(n,m,\gamma)\in (0,1)\), then we have \(\mathrm{spt}\|V\| \cap C_\gamma \subset B_{(1+\gamma)/2}\), and hence we can apply 7 in \(B_{(1+\gamma)/2}\) to get \[d_P(x)^2 \leq c(n, m, \gamma) \int_{B_1} d_P^2\, d\|V\| \leq c E_{V, P}(0, 1).\] On the other hand, if \(d_P(x) \geq \eta\), then in the ball \(B_{\eta/2}(x)\) we have \(d_P \geq d_P(X)/2\) and by monotonicity \(\|V\|(B_{\eta/2}(X)) \geq \omega_n (\eta/2)^n\), and so we can estimate \[(d_P(x)/2)^2 \omega_n (\eta/2)^n \leq \int_{B_{\eta/2}} d_P^2\, d\|V\| \leq E_{V, P}(0, 1). \qedhere\] ◻
Lemma 3 (\(L^2\) height controls tilt). Let \(V\) be a stationary rectifiable \(n\)-varifold in \(B_1(0)\). Then for any \(z \in \mathbb{R}^{n+m}\) and \(\gamma \in (0, 1)\) we have \[\int_{B_\gamma} \|\pi_T - \pi_P\|^2\, d\|V\| \leq c(n, \gamma) \int_{B_1} d_{z + P}^2\, d\|V\|.\]
Proof. The proof is standard, see e.g. [36]. ◻
The following lemma is the first relationship which relies on being area-minimizing.
Lemma 4 (\(L^2\) controls mass excess [25]). Let \(T\) be an area-minimizing integral \(n\)-current in \(C_1(0)\) satisfying \[\label{eqn:mass-by-L2-hyp} (\partial T) \llcorner C_1(0) = 0, \quad \sup_{x \in \mathrm{spt}\|T\| \cap C_1(0)} d_P(x) \leq 1.\tag{9}\] Then for any \(\gamma \in (0, 1)\) we have \[\label{eqn:mass-by-L2-concl1} \mathcal{E}_{T, P}(0, \gamma) \leq c(n, m, \gamma) E_{T, P}(0, 1) .\tag{10}\]
Lemma 5 (Zero projection implies zero mass). Let \(T\) be an area-minimizing integral \(n\)-current in \(C_1(0)\) satisfying \[\begin{gather} \label{eqn:zero-mass-hyp} (\partial T) \mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_1(0) = 0, \quad \Theta_T(0, 1) \leq \theta + \epsilon, \quad E_{T, P}(0, 1) \leq \epsilon^2. \end{gather}\tag{11}\] Then, provided \(\epsilon= \epsilon(n, m, \theta)\) is sufficiently small, we have \[\label{eqn:zero-mass-concl} \pi_{P \#} (T \mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_1) = q\llbracket B^n_1(0) \rrbracket \text{ for some |q| \leq \theta.}\tag{12}\] Moreover, if \(q = 0\), then for any \(\gamma \in (0, 1)\) we have \(T \mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_\gamma(0) = 0\) provided \(\epsilon= \epsilon(n, m, \theta, \gamma)\) is sufficiently small.
Proof. By the constancy theorem for currents we know that \(\pi_{P\#}(T\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_1) = q\llbracket B^n_1(0)\rrbracket\) for some integer \(q\in \mathbb{Z}\). Fix any \(\gamma\in (0,1)\). We know that, up to changing the orientation on \(P\), \[\gamma^{-n}\|T\|(C_\gamma) - \gamma^{-n}|q|\omega_n = \mathcal{E}_{T,P}(0,\gamma) \geq 0\] and so \(|q|\omega_n \leq \|T\|(C_\gamma)\). But by Lemma 2 and hypothesis 11 , provided \(\epsilon(n, m, \gamma)\) is sufficiently small, we have \[\label{eqn:zero-1} \|T\|(C_\gamma) \leq \|T\|(B_{\gamma+c\epsilon}) \leq (\gamma+c\epsilon)^n\Theta_T(0,1)\omega_n \leq \theta\omega_n + c(n,m,\theta)\epsilon.\tag{13}\] and so combining these inequalities we get that \(|q|\leq \theta + c(n,m,\theta)\epsilon\), and hence as \(q\in \mathbb{Z}\) if we choose \(\epsilon=\epsilon(n,m,\theta)\) sufficiently small we get \(|q|\leq\theta\).
For the second claim we argue by contradiction. Suppose, for a fixed \(\gamma \in (0, 1)\), there are sequences \(\epsilon_i \to 0\), \(T_i\) satisfying the hypotheses 11 (with \(\epsilon_i, T_i\) in place of \(\epsilon, T\)) and \[\pi_{P\#}(T_i \mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_1) = 0,\] but for which \(T_i \mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_\gamma \neq 0\).
From 13 , we know that for any \(\gamma' \in (0, 1)\) we have uniform mass bounds \(\|T_i\|(C_{\gamma'}) \leq c(n, m, \theta)\) for all \(i \gg 1\), and therefore after passing to a subsequence we can find an area-minimizing integral \(n\)-current \(T'\) (without boundary) in \(C_1\) so that \(T_i \to T'\) as currents in \(C_1\), \(|T_i| \to |T|\) as varifolds in \(C_1\), and \(\mathrm{spt}\|T_i\| \to \mathrm{spt}\|T'\|\) in the local Hausdorff distance in \(C_1\). But then \(T'\) must satisfy \[E_{T', P}(0, 1) = 0, \quad \pi_{P\#}(T' \mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_1) = 0,\] which implies \(T' \mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_1 = 0\). But now from our contradiction hypotheses and Lemma 2 we can find \(x_i \in \mathrm{spt}\|T_i\| \cap C_\gamma\) with \(d_P(x_i) \to 0\), which means by the local Hausdorff convergence of supports there must be some \(x \in \mathrm{spt}\|T'\| \cap C_{(1+\gamma)/2}\), which is a contradiction. ◻
We also require the following improved \(L^2-L^\infty\) estimate for area-minimizers.
Theorem 3 (Improved \(L^2-L^\infty\) estimate). Let \(Q \in \mathbb{N}\), \(P = \mathbb{R}^n \times \{0^m\}\), \(p_1, \ldots, p_N \in \mathbb{R}^m\), and write \(\boldsymbol{\pi} = \cup_{i=1}^N (p_i + P)\) for the union of parallel planes. Let \(T\) be an area-minimizing integral \(n\)-current in \(C_1(0)\) satisfying \[\|T\|(C_1(0)) \leq (Q+\tfrac{1}{2})\omega_n, \quad \pi_{P\#}(T \mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_1(0)) = Q\llbracket B_1^n \rrbracket,\] \[E := \int_{C_1} d(y, \boldsymbol{\pi})^2\, d\|T\| \leq 1.\] Then for any \(\gamma < 1\), we have \[\mathrm{spt}\|T\| \cap C_\gamma(0) \subset \{ x : d(x, \boldsymbol{\pi}) \leq c E^{1/2} \}.\] for \(c = c(n, m, Q, N, \gamma) > 0\).
In this section we recall some standard facts about multi-valued functions which minimize the Dirichlet energy. Standard references are [32]–[34].
We write \(\mathcal{A}_Q(\mathbb{R}^m)\) for the space of unordered \(Q\)-tuples, identified by sums of \(Q\) Dirac point masses in \(\mathbb{R}^m\). We write \(\mathcal{G}\) for the corresponding Almgren metric on \(\mathcal{A}_Q(\mathbb{R}^m)\). A \(W^{1,2}\) function \(v:B^n_1\to \mathcal{A}_Q(\mathbb{R}^m)\) is Dirichlet-minimizing if \[\int_{B_1} |Dv|^2 \leq \int_{B_1} |Du|^2\] for any \(W^{1,2}\) function \(u:B_1\to \mathcal{A}_Q(\mathbb{R}^m)\) which obeys \(u = v\) \(\mathcal{L}^n\)-a.e. on \(B_1\setminus\Omega\) for some open \(\Omega\subset\subset B^n_1\). It is a standard fact (see e.g. [33]) that Dirichlet-minimizers are \(C^{0,\alpha}\) Hölder continuous for some universal \(\alpha = \alpha(n,Q)\in (0,1)\).
If \(v:B^n_1\to \mathcal{A}_Q(\mathbb{R}^m)\) is Dirichlet-minimizing, we call \(x\in B_1\) a regular point if there exists an open neighborhood \(U_x\subset B_1\) of \(x\) such that \(v|_{U_x} = \sum^Q_{j=1}\llbracket v_j\rrbracket\) for some analytic functions \(v_1,\dotsc,v_Q:U_x\to \mathbb{R}^m\) with either \(v_i\equiv v_j\) or \(v_i(y)\neq v_j(y)\) for all \(y\in U_x\). We write \(\mathrm{reg}(v)\) for the set of regular points of \(v\), with the singular set then \(\mathrm{sing}(v):= B_1\setminus \mathrm{reg}(v)\).
An important fact we will need is the following dimension bound on the singular set of a Dirichlet-minimizer.
Theorem 4. Suppose \(v\in W^{1,2}(B^n_1;\mathcal{A}_Q(\mathbb{R}^m))\) is Dirichlet-minimizing. Then \(\mathrm{sing}(v)\) has Hausdorff dimension \(\leq n-2\). Moreover, if \(n=2\), \(\mathrm{sing}(v)\) consists of isolated points.
Proof. See [33]. ◻
The standard tool for analyzing a (non-zero) Dirichlet-minimizer \(v:\Omega\to\mathcal{A}_Q(\mathbb{R}^m)\) in an open domain \(\Omega \subset \mathbb{R}^n\) is the frequency function \(N\). We give a definition following [25]. Fix \(\phi:[0,\infty)\to \mathbb{R}\) a non-increasing Lipschitz function satisfying \(\phi(s)\equiv 1\) for \(s\in [0,1/2]\) and \(\phi(s)\equiv 0\) for \(s\in [1,\infty)\). For each \(z\in \Omega\) and \(\rho\in (0,\mathrm{dist}(z,\partial\Omega))\), define \[D_{v,z}(\rho):=\rho^{2-n}\int|Dv|^2\phi(r/\rho) \qquad \text{and} \qquad H_{v,z}(\rho):= -\rho^{1-n}\int|v|^2\frac{1}{r}\phi^\prime(r/\rho),\] where \(r = r(x):= |x-z|\). The frequency function is then \[N_{v,z}(\rho):= \frac{D_{v,z}(\rho)}{H_{v,z}(\rho)}.\] We note that letting \(\phi\) decrease to the indicator function of the interval \([0,1]\), one recovers the “usual” frequency function as originally defined by Almgren. The key property of the frequency function is that \(N_{v,z}(\rho)\) is monotone non-decreasing in \(\rho\) for each \(z\), and furthermore \(N_{v,z}(\rho)\) is constant in \(\rho\) if and only if \(v\) is homogeneous of degree \(N\equiv N_{v,z}(\rho)\) centered at \(z\). If \(0 \in \Omega\), we will often use the shorthand \(N_v(\rho) \equiv N_{v, 0}(\rho)\).
Multi-valued Dirichlet-minimizing functions are important because of Almgren’s (strong) Lipschitz approximation theorem for area-minimizing currents. Here we write \(\mathbf{v}(u)\) for the natural current associated to the graph of a Lipschitz multi-valued function \(u: B^n_r\to \mathcal{A}_Q(\mathbb{R}^m)\).
Theorem 5. Fix \(Q\in \{1,2,\dotsc\}\), \(P = \mathbb{R}^n \times \{0^m\}\), and let \(T\) be an area-minimizing integral \(n\)-current in \(C_{1}(0) \subset \mathbb{R}^{n+m}\) satisfying \[\begin{gather} (\partial T)\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_1 = 0, \quad \sup_{x\in\mathrm{spt}\|T\|}d_{P}(x)<\infty, \quad \pi_{P\#}T = Q\llbracket B_1^n\rrbracket, \\ \mathcal{E}:=\mathcal{E}_{V,P}(0,1)<\epsilon^2. \end{gather}\] Then provided \(\epsilon(n, m, Q) > 0\) is sufficiently small, there exists a Lipschitz function \(u:B^n_{1/2}\to \mathcal{A}_Q(\mathbb{R}^m)\) and a closed set \(K\subseteq B^n_{1/2}\) so that
\(T \mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(K\times \mathbb{R}^m) = \mathbf{v}(u) \mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(K\times \mathbb{R}^m)\);
\(\textrm{Lip}(u) \leq c\mathcal{E}^{\gamma}\);
\(\mathcal{H}^n(B^n_{1/2}\setminus K) + \|T\|((B^n_{1/2}\setminus K)\times\mathbb{R}^m) \leq c\mathcal{E}^{1+\gamma}\).
Here \(\gamma = \gamma(n,m,Q)\), \(c = c(n, m, Q)\) are positive constants.
Proof. See e.g. [25]. ◻
A crucial tool for us is that at large scales a \(2\)-dimensional area-minimizing current \(T\) with quadratic area growth looks close to a fixed tangent cone (which is necessarily a union of transverse planes with multiplicity), and moreover \(T\) decays to this tangent cone at infinity at a controlled rate. This is the content of the following.
Theorem 6 (Uniqueness of tangent cones at infinity, [24]). Let \(T\) be an area-minimizing integral \(2\)-current in \(\mathbb{R}^{2+m}\) with \(\partial T = 0\) and \(\Theta := \Theta_T(\infty) < \infty\). Then there are transverse \(2\)-planes \(\{ P_1, \ldots, P_\ell \}\) and multiplicities \(n_1, \ldots, n_\ell \in \mathbb{N}\) so that \(\mathbf{C}= n_1 \llbracket P_1 \rrbracket + \cdots + n_\ell \llbracket P_\ell \rrbracket\) is the unique tangent cone of \(T\) at infinity. In other words, \((1/R)_\# T \to \mathbf{C}\) as currents and varifolds, and \((1/R) \mathrm{spt}\|T\| \to \mathrm{spt}\|\mathbf{C}\|\equiv \cup_i P_i\) locally in the Hausdorff distance, as \(R\to\infty\).
Moreover, there is an \(\alpha = \alpha(m, \Theta) > 0\) so that for all \(R \gg 1\) we have the scale-invariant decay: \[\begin{gather} R^{-1} d_H( \mathrm{spt}\|T\| \cap B_R, \mathrm{spt}\|\mathbf{C}\| \cap B_R) \leq R^{-\alpha}, \tag{14} \\ R^{-n-2} \int_{B_R} \mathrm{dist}(x, \mathbf{C})^2\, d\|V\|(x) \leq R^{-\alpha}. \tag{15} \end{gather}\]
Proof. This follows primarily from the reverse epiperimetric inequality shown in [24]. Specifically, [24] implies there is an \(\epsilon= \epsilon(m, \Theta) \in (0, 1)\) and \(R_0 = R_0(T) \gg 1\) so that \[\label{eqn:unique-1} \Theta - \Theta_T(0, R) \leq (R_0/R)^{2\epsilon}(\Theta - \Theta_T(0, R_0)) \quad \forall R \geq R_0.\tag{16}\] Arguing as in [39], 16 implies that given any sequence \(R_i \to \infty\), the currents \(\partial(((1/R_i)_\# T) \llcorner B_1)\) form a Cauchy sequence in the flat norm, which implies \((1/R)_\# T \to \mathbf{C}\) as currents (for some area-minimizing cone \(\mathbf{C}\)), and hence as varifolds by the minimizing property (cf. Lemma 4 and Theorem 5) and in support by the monotonicity formula.
To see the convergence rates 14 and 15 , we can use Lemma 11. For any \(\delta>0\), by enlarging \(R_0\) one can assume that \(d_H( \mathrm{spt}\|T\| \cap B_R, \mathrm{spt}\|\mathbf{C}\| \cap B_R) \leq \delta R\) for all \(R \geq R_0\). By picking \(\delta = \delta(\mathbf{C})\) suitably small, we can find a smooth, \(1\)-homogeneous function \(f\) on \(\mathbb{R}^{2+m} \setminus \{0\}\) which agrees with the signed distance to \(\mathbf{C}\) in the region \[\{ x \in \mathbb{R}^{2+m} \setminus \{0\} : \mathrm{dist}(x, \mathbf{C}) \leq 2\delta |x| \}.\] Applying Lemma 11 to \(f\) with the decay 16 , we get \[\label{eqn:unique-2} |e(R) - e(S)| \leq c(m, \Theta) (\log(R/S) + 1) (R_0/S)^{\epsilon} \quad \forall R_0 < S < R,\tag{17}\] where we write \(e(R) := R^{-n-2} \int_{A_{R, R/2}} \mathrm{dist}(x, \mathbf{C})^2\, d\|V\|\). Taking \(R_i = R_0 2^i\) we deduce from 17 that \[e(R_i) = e(R_i) - \lim_{R \to \infty} e(R) \leq \sum_{j \geq i} |e(R_{j+1}) - e(R_j)| \leq c(m, \Theta) R_i^{-\epsilon},\] and hence \[\begin{align} \int_{B_{R_i}} \mathrm{dist}(x, \mathbf{C})^2 d\|V\| &\leq \int_{B_{R_0}} \mathrm{dist}(x, \mathbf{C})^2 d\|V\| + \sum_{j=1}^i e(R_j)R_j^{n+2} \nonumber \\ &\leq \int_{B_{R_0}} \mathrm{dist}(x, \mathbf{C})^2 d\|V\| + c R_i^{n+2-\epsilon} \nonumber \\ &\leq c(m, \Theta) R_i^{n+2-\epsilon} \label{eqn:unique-3} \end{align}\tag{18}\] provided \(i \gg 1\). Combing the above with 17 and ensuring \(R \gg 1\) we get 14 with \(\alpha = \epsilon/2\). The decay estimate in 14 then follows from 15 and Lemma 2. ◻
In this section we give a generalized version of Krummel–Wickramasekera’s planar frequency monotonicity formula for area-minimizing \(n\)-currents in \(\mathbb{R}^{n+m}\), as originally introduced in [25]. For this, we fix \(\phi:[0,\infty)\to \mathbb{R}\) the Lipschitz function with \(\phi(s)\equiv 1\) for \(s\in [0,1/2]\), \(\phi(s) = 2-2s\) for \(s\in [1/2,1]\), and \(\phi(s)\equiv 0\) for \(s\in [1,\infty)\). We also write \(r(x):= |\pi_P(x)|\) for the cylindrical radius with respect to the plane \(P := \mathbb{R}^n\times\{0^m\}\). Then define the quantities \[H_{T,P}(\rho):= -\rho^{1-n}\int d^2_P|\nabla r|^2\frac{1}{r}\phi^\prime(r/\rho)\, d\|T\|,\] \[D_{T,P}(\rho) := \frac{1}{2}\rho^{2-n}\int\|\pi_T-\pi_P\|^2\phi(r/\rho)\, d\|T\|,\] and, when \(H_{T,P}(\rho)>0\), \[N_{T,P}(\rho):= \frac{D_{T,P}(\rho)}{H_{T,P}(\rho)}.\] The function \(N_{T,P}\) is an intrinsic notion of frequency for the current \(T\) relative to the plane \(P\) (at scale \(\rho\)), which effectively reduces to the ‘standard’ frequency \(N_v\) if \(T\) is represented by the multi-graph of a ‘small’ function \(v\) over \(P\).
The main theorem of this section is the following. The proof is a refinement of the calculations in the proof of the planar frequency monotonicity given in [25], which we defer to Appendix 8.
Theorem 7. Let \(\theta\geq 1\) and \(P = \mathbb{R}^n\times\{0^m\}\). Then, there exists \(\delta = \delta(n,m,\theta)\in (0,1)\) such that if \(T\) is an area-minimizing integral \(n\)-current in \(C_{2\rho}(0)\) obeying \[(\partial T)\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_{2\rho}(0) = 0,\qquad \Theta_T(0,\rho/2)\geq 1/2, \qquad \Theta_{T}(0,2\rho)\leq\theta,\] \[E_{T,P}(0,2\rho)\leq E \leq \delta^2,\] and such that \(\mathrm{spt}\|T\|\) does not coincide with \(P\) in \(C_{2\rho}(0)\), then \[\label{eqn:freq-mono-concl1} \frac{d}{d\rho} \log N_{T, P}(\rho) \geq -c \rho^{-1} E^{\gamma} - c' E^{\gamma-\gamma_1} \frac{d}{d\rho}\left[ (\rho^{-2} D_{T, P}(\rho))^{\gamma_1} \right],\tag{19}\] and \[\label{eqn:freq-mono-concl2} \left| \frac{d}{d\rho} \log H_{T, P}(\rho) - 2 \rho^{-1} N_{T, P}(\rho) \right| \leq c \rho^{-1} E^{\gamma}.\tag{20}\] Here, \(0<c^\prime<c\) and \(0<\gamma_1<\gamma\) are all constants depending only on \(m,n,\theta\).
The primary corollary of Theorem 7 for us is almost-monotonicity of \(N_{T,P}\) at large scales when \(T\) decay polynomially to \(P\) at infinity (almost-monotonicity of \(N_{T,P}\) at small scales given polynomial decay to \(P\) at a point was the original theorem of [25]). We remark that the differential inequality is more general, and would also imply an almost-monotonicity of \(N_{T,P}\) if \(E_{T,P}(\rho)\) were to decay logarithmically in \(\rho\).
Corollary 1. Let \(T\) be an area-minimizing integral \(n\)-current in \(B_R\subset\mathbb{R}^{n+m}\) whose support does not coincide with a plane. Suppose that \[(\partial T)\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}B_R = 0,\qquad \Theta_T(0,1/2)\geq 1/2,\qquad \Theta_T(0,R)\leq \theta,\] and, for some fixed \(\eta,\alpha>0\), \[\rho^{-n-2}\int_{B_\rho}d_P^2\, d\|T\| \leq \eta^2\rho^{-2\alpha} \qquad \text{for all }1\leq\rho\leq R.\] Then, provided \(\eta = \eta(n,m,\theta)\in (0,1)\) is sufficiently small, there are constants \(c^\prime<c\), \(\gamma_1<\gamma\) depending only on \(n,m,\theta,\alpha\) such that the function \[\begin{gather} \label{eqn:cor1-freq-mono-concl1} \rho\mapsto N_{T,P}(\rho) \exp(-c\eta^{2\gamma}\rho^{-2\alpha\gamma}+c^\prime\eta^{2\gamma-2\gamma_1}(\rho^{-2}D_{T,P}(\rho))^{\gamma_1}) \end{gather}\qquad{(1)}\] is non-decreasing for \(1\leq\rho\leq R/4\).
Proof. This follows directly from 19 , since by Lemma 2 we have \(E_{T,P}(\rho)\leq \eta^2\rho^{-2\alpha}\) for all \(1\leq\rho\leq R/2\), provided \(\eta(n, m, \theta)\) is small. From Theorem 7 we get \[\frac{d}{d\rho}\log N_{T,P}(\rho) \geq -c\eta^{2\gamma}\rho^{-2\alpha\gamma-1} - c^\prime \eta^{2\gamma-2\gamma_1}\frac{d}{d\rho}\left[(\rho^{-2}D_{T,P}(\rho))^{\gamma_1}\right]\] for all \(1 \leq \rho \leq R/4\), which implies ?? is monotone. ◻
The second corollary is the standard growth and doubling estimates which follow from the almost monotonicity. We also defer the proof to Appendix 8.
Corollary 1. Let \(T\) be an area-minimizing integral \(n\)-current in \(C_{2\rho_2}\), for \(0<\rho_1<\rho_2\), and suppose that \[(\partial T)\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_{2\rho_2} = 0,\qquad \Theta_T(0,\rho_1/2)\geq 1/2,\qquad \Theta_T(0,2\rho_2)\leq\theta,\] and \[E_{T,P}(2\rho)\leq \epsilon^2 \quad \text{and} \quad N_1\leq N_{T,P}(\rho)\leq N_2\qquad \text{for all }\rho_1\leq\rho\leq\rho_2.\] Then, provided \(\epsilon= \epsilon(m,n,\theta)\) is sufficiently small, we have \[\label{eqn:H-control-concl1} (\rho_2/\rho)^{2N_1 - c \epsilon^{2\gamma}} \leq \frac{H_{T, P}(\rho_2)}{H_{T, P}(\rho_1)} \leq (\rho_2/\rho_1)^{2N_2 + c \epsilon^{2\gamma}}\qquad{(2)}\] and \[\label{eqn:H-control-concl2} E_{T, P}(\rho_2) \leq c (\rho_2/\rho_1)^{2N_2 + c \epsilon^{2\gamma}} E_{T, P}(\rho_1).\qquad{(3)}\] where \(c = c(n,m,\theta)\in (0,\infty)\).
In this section we prove an asymptotic expansion for \(u:Q\mathbb{R}^2\to \mathbb{R}^m\) solving the minimal surface system. We will make use of the Fourier expansion for functions \(u:(a,b)\times QS^1\to \mathbb{R}\). Indeed, for any such function, and for each \(r\), we can expand \(u(r,\cdot)\) in terms of the standard orthogonal basis of \(L^2(QS^1)\): \[u(r,\theta) = a_0(r) + \sum^\infty_{k=1} \big(a_k(r)\cos(k\theta/Q) + b_k(r)\sin(k\theta/Q)\big),\] where \[a_0(r) = \frac{1}{2\pi Q}\int^{2\pi Q}_0 u(r,\theta)\, d\theta, \quad a_k(r) = \frac{1}{\pi Q}\int^{2\pi Q}_0 u(r,\theta)\cos(k\theta/Q)\, d\theta,\] \[b_k(r) = \frac{1}{\pi Q}\int^{2\pi Q}_0u(r,\theta)\sin(k\theta/Q)\, d\theta.\] Thus, \[\int^{2\pi Q}_0u(r,\theta)^2\, d\theta = a_0(r)^2+ \sum_{k\geq 1}\left(a_k(r)^2+b_k(r)^2\right).\] We note that if \(u:(1,\infty)\times QS^1\to \mathbb{R}^m\) and \(u = O(r^{\alpha+})\), then from the above expressions we see that \(a_k(r) = O(r^{\alpha+})\) and \(b_k(r) = O(r^{\alpha+})\) for each \(k\) also.
If \(\Delta u = 0\), then by considering the corresponding ODEs solved by \(a_k(r), b_k(r)\) we may expand \[\begin{align} u = A_0 + B_0\log(r) + \sum^\infty_{k=1}&\left(A_kr^{k/Q}+A_{-k}r^{-k/Q}\right)\cos(k\theta/Q)\nonumber\\ & + \sum_{k=1}^\infty\left(B_kr^{k/Q}+B_{-k}r^{-k/Q}\right)\sin(k\theta/Q), \label{eqn:q-harmonic-fourier} \end{align}\tag{21}\] for constants \(A_k,B_k\in \mathbb{R}^m\) so that, for each \(r\), \[\begin{align} \int_{QS^1}u(r,\theta)^2\, d\theta = (A_0+B_0\log(r))^2+\sum^\infty_{k=1}&\left(A_kr^{k/Q}+A_{-k}r^{-k/Q}\right)^2\nonumber\\ & + \sum_{k=1}^\infty \left(B_k r^{k/Q}+B_{-k}r^{-k/Q}\right)^2. \label{eqn:q-harmonic-L2} \end{align}\tag{22}\] From 22 it follows that if \(u\) is harmonic and \(u = O(r^{\alpha+})\) then \(A_k,B_k=0\) for \(k/Q>\alpha\). Conversely, if \(A_k = B_k =0\) for \(k>k_0\), then \(u = O(r^{k_0/Q+})\).
Our first lemma extends the above analysis to certain Poisson equations.
Lemma 6. Take \(Q\in \mathbb{N}\), and \(\alpha>0\). Suppose \(f:[1,\infty)\times QS^1\to \mathbb{R}^m\) is a smooth function satisfying \(f = O(r^{-1-\alpha+})\). Then, there is a smooth function \(u:(1,\infty)\times QS^1\to \mathbb{R}^m\) solving \[\Delta u = f \quad \text{with} \quad u = O_{1,1-}(r^{1-\alpha+}).\] In fact if \(f = O_{k,\gamma}(r^{-1-\alpha+})\) for some \(\gamma\in (0,1)\), then \(u = O_{k+2,\gamma}(r^{1-\alpha+})\).
Proof. We shall construct a solution in terms of its complex Fourier expansion \[\label{eqn:ps-1} u(r,\theta) = \sum_{k\in\mathbb{Z}}a_k(r)e^{\sqrt{-1}k\theta/Q},\tag{23}\] for \(a_k(r)\) to be defined below. Set \[f_k(r):=\frac{1}{2\pi Q}\int_0^{2\pi Q}f(r,\theta)e^{-\sqrt{-1}k\theta/Q}\, d\theta ,\] and note that \(f_k = O(r^{-1-\alpha+})\). (We will slightly abuse notation here as \(a_k(r),f_k(r)\in\mathbb{C}^m\).)
We then have \(\Delta u = f\) only if each coefficient \(a_k\) solves the ODE \[a_k^{\prime\prime}+r^{-1}a_k^\prime - r^{-2}(k/Q)^2a_k = f_k,\] which can be written in divergence form as \[\label{eqn:ps-2} r^{k/Q-1}\big(r^{1-2k/Q}(r^{k/Q}a_k)^\prime\big)^\prime = f_k.\tag{24}\] For any \(k\geq 0\), since \(\alpha>0\) we can find explicit solutions \(a_k(r)\) of 24 of the form \[a_k(r) = r^{-k/Q}\int^r_1\left(s^{2k/Q-1}\int^s_{s_k}t^{1-k/Q}f_k(t)\, dt\right)\, ds,\] where we set \(s_k=\infty\) if \(k/Q>1-\alpha\) and \(s_k=1\) if \(k/Q\leq 1-\alpha\).
We now claim that these choices of \(a_k\) in 23 give the desired solution. First, let us fix \(\beta\in (0,\alpha)\) so that \(\{k\in \mathbb{Z}:k/Q>1-\alpha\} = \{k\in \mathbb{Z}:k/Q>1-\beta\}\). Then, observe that by our assumption of \(f = O(r^{-1-\alpha+})\), for \(\epsilon= \epsilon(\alpha,\beta)\in (0,1)\) sufficiently small, we have \[{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert f \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_\beta^2:=\int^\infty_1 t^{1+2\beta}\|f(t,\cdot)\|^2_{L^2(QS^1)}\, dt \leq c(f,\epsilon)\int^\infty_1 t^{-1+2(\beta-\alpha+\epsilon)}\, dt<\infty.\] We now compute bounds on \(a_k(r)\) as follows. If \(k/Q>1-\alpha\) (and hence \(k/Q>1-\beta\)) and \(r\geq 1\), we have: \[\begin{align} |a_k(r)| & \leq r^{-k/Q}\int^r_1\left( s^{2k/Q-1}\int^\infty_1 t^{1-k/Q}|f_k(t)|\, dt\right)\, ds\\ & \leq r^{-k/Q}\int^r_1 s^{2k/Q-1}\left(\int^\infty_1 t^{1+2\beta}f_k(t)^2\, dt\right)^{1/2}\left(\int^\infty_s t^{1-2k/Q-2\beta}\, dt\right)^{1/2}\, ds\\ & \leq \left(\int^\infty_1 t^{1+2\beta}f_k(t)^2\, dt \right)^{1/2}\cdot r^{-k/Q}\int^r_1 s^{2k/Q-1}\left(\frac{s^{2-2k/Q-2\beta}}{2k/Q-2+2\beta}\right)^{1/2}\, ds\\ & \leq \frac{1}{\sqrt{2}}\left(\int^\infty_1 t^{1+2\beta}f_k(t)^2\, dt \right)^{1/2}\frac{r^{1-\beta}-r^{-k/Q}}{(k/Q+(1-\beta))(k/Q-(1-\beta))^{1/2}}. \end{align}\] The above computation was justified because \(2-2k/Q-2\beta<0\). We deduce that for every such \(k\), we have \[\label{eqn:ps-3465} |a_k(r)|^2 \leq c(\beta,\alpha,Q)r^{2-2\beta}\int^\infty_1 t^{1+2\beta}f_k(t)^2\, dt \qquad \text{for all }r\geq 1.\tag{25}\] On the other hand, if \(k/Q\leq 1-\alpha\), then we can compute similarly \[\begin{align} |a_k(r)| & \leq r^{-k/Q}\int^r_1 s^{2k/Q-1}\left(\int^\infty_1 t^{1+2\beta}f_k(t)^2\, dt\right)^{1/2}\left(\int^s_1 t^{1-2k/Q-2\beta}\, dt\right)^{1/2}\, ds\\ & \leq \left(\int^\infty_1 t^{1+2\beta}f_k(t)^2\, dt\right)^{1/2} r^{-k/Q}\int^r_1 s^{2k/Q-1}\left(\frac{s^{2-2k/Q-2\beta}-1}{2-2k/Q-2\beta}\right)^{1/2}\, ds\\ & \leq \frac{1}{\sqrt{2}}\left(\int^\infty_1 t^{1+2\beta}f_k(t)^2\, dt\right)^{1/2}\frac{r^{1-\beta}-r^{-k/Q}}{(k/Q+(1-\beta))(1-\beta-k/Q)^{1/2}}, \end{align}\] which is justified because \(1-k/Q-\beta \geq \alpha-\beta>0\). We deduce that 25 also holds for every such \(k\), and therefore \[\sum_k |a_k(r)|^2 \leq c(\alpha,\beta,Q)r^{2-2\beta}\int^\infty_1 t^{1+2\beta}\|f(t,\cdot)\|_{L^2(QS^1)}^2\, dt \quad\text{for all }r\geq 1.\] It follows that the sum in 23 converges in \(L^2\) to an \(L^2\) function \(u:(1,\infty)\times QS^1\to \mathbb{R}^m\) which weakly solves the equation \(\Delta u = f\) and admits the \(L^2\) bound \[\label{eqn:ps-4} \|u(r,\cdot)\|_{L^2(QS^1)}\leq c(\alpha,\beta,Q)r^{1-\beta}{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert f \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_\beta \quad \text{for all }r\geq 1\text{ and }0<\beta<\alpha.\tag{26}\] From standard elliptic PDE theory, \(u\) is smooth and solves \(\Delta u = f\) in the classical sense. From 26 we have \(u = O(r^{1-\alpha+})\), and so the asymptotic \(u = O_{1,1-}(r^{1-\alpha+})\) follows from \(C^{1,\alpha}\) Schauder theory. The further asymptotics \(u = O_{k+2,\gamma}(r^{1-\alpha+})\) when \(f = O_{k,\gamma}(r^{-1-\alpha+})\) follow from \(C^{2,\alpha}\) Schauder theory. ◻
The next two lemmas are the main results of this section.
Lemma 7 (Refined asymptotics near a multiplicity \(Q\) plane). Take \(Q \in \mathbb{N}\), \(\alpha \in (0, 1]\). Suppose \(v:(1,\infty)\times QS^1\to \mathbb{R}^m\) is a \(C^2\) function solving the minimal surface system 4 which admits the decay \(v = O(r^{1-\alpha+})\). Let \(\alpha_Q:= \max\{k/Q: k/Q\leq1-\alpha,\;k\in\mathbb{Z}\}\).
Then, we can find \(A_k,B_k\in \mathbb{R}^m\) and \(r_0>1\) so that, for \(r\geq r_0\), \(v\) admits the expansion \[\begin{align} v(r,\theta) = & \sum_{0<k/Q\leq\alpha_Q}\big(A_kr^{k/Q}\cos(k\theta/Q) + B_kr^{k/Q}\sin(k\theta/Q)\big) \label{eqn:refined-concl1}\\ &+ A_0\log(r) + B_0 \nonumber \\ &+ \sum_{3\alpha_Q-2<k/Q<0}\big(A_kr^{k/Q}\cos(k\theta/Q)+B_kr^{k/Q}\sin(k\theta/Q)\big) \nonumber \\ &+ O_{2,1-}(r^{3\alpha_Q-2+}). \nonumber \end{align}\tag{27}\] In particular, if \(Q=2\) we have the expansion \[v(r,\theta) = A_1 r^{1/2}\cos(\theta/2) + B_1r^{1/2}\sin(\theta/2) + A_0\log(r) + B_0 + O_2(r^{-1/2+}),\] and if \(Q=1\) we have the expansion \[v(r,\theta) = A_0\log(r) + B_0 + \frac{A_{-1}}{r}\cos(\theta) + \frac{B_{-1}}{r}\sin(\theta) + O_2(r^{-2+}).\]
Proof. Let \(\alpha_Q\) be the smallest number \(\in [0,1-\alpha]\) for which \(v = O(r^{\alpha_Q+})\). If \(r\gg 1\), we can assume that \(|v(r,\theta)|/r\leq\eta\), for some \(\eta = \eta(n,m)\) suitably small. By applying Allard’s regularity theorem to the single-sheeting surfaces \(\{(r\cos(\theta),r\sin(\theta),v(r,\theta): r>0,\;|\theta-\theta_0|<\pi/2\}\) for suitable choices of \(\theta_0\), we get the a priori bounds \[\begin{align} |Dv|_{C^{0}(\{R<r<2R\})} + R^\gamma[Dv]_{\gamma;\{R<r<2R\}} & \leq c(m,\gamma)R^{-1}|v|_{C^0(\{R/2<r<4R\})}\\ & \leq c(m,\gamma,\epsilon)\eta R^{-1+\alpha_Q+\epsilon} \end{align}\] for any \(\gamma\in (0,1)\), any \(\epsilon>0\), and all \(R\gg 1\). We can then apply Schauder theory to the minimal surface system 4 to get \[R^{k-1}|D^kv|_{C^0(\{R<r<2R\})} \leq c(m,\alpha,\epsilon,k)R^{-1+\alpha_Q+\epsilon} \qquad \text{for all }k\geq 2.\] In other words, \(v = O_k(r^{\alpha_Q+})\) for all \(k\geq 2\).
Using form 5 of the minimal surface system, \(v\) solves \[\Delta v = F(Dv,D^2v) \equiv Dv\star Dv\star D^2v = O_1(r^{3\alpha_Q-4+}).\] From Lemma 6, on \(\{r>r_0\}\) for \(r_0\) large there is a solution \(v_p\) to \[\Delta v_p = F(Dv,D^2v) \quad \text{which obeys} \quad v_p = O_{2,1-}(r^{3\alpha_Q-2+}).\] Now \(v-v_p\) is a harmonic function which is \(O(r^{\alpha_Q+})\), and hence by the discussion below 22 \(v-v_p = O_{2,1-}(r^{k_0/Q+})\) for \(k_0\) being the maximal integer satisfying \(k_0/Q\leq \alpha_Q\). We deduce that \(v = O(r^{\max\{k_0/Q,3\alpha_Q-2\}+})\), and since \(3\alpha_Q - 2 < \alpha_Q\), by our choice of \(\alpha_Q \in [0, 1)\) as the minimal exponent for which \(v = O(r^{\alpha_Q+})\) we must have \(\alpha_Q = k_0/Q\). The expansion 27 then follows from the Fourier expansion 21 for \(v - v_p\). ◻
An important and immediate consequence of Lemma 7 is that if \(Q = 2\) then \(v\) grows at most logarithmically outside the \(4\)-dimensional subspace \(\mathbb{R}^2 \oplus A_1 \mathbb{R}\oplus B_1 \mathbb{R}\). The below Lemma generalizes this principle to arbitrary \(Q\).
Lemma 8 (Weak subspace rigidity near a planar end). Take \(Q \in \mathbb{N}\), and \(\alpha > 0\). Let \(v: (1,\infty)\times QS^1\to \mathbb{R}^m\) be a smooth solution to the minimal surface system 4 satisfying \(v = O(r^{1-\alpha+})\). Let \(k:=\max\{k\in \mathbb{Z}:k/Q\leq 1-\alpha\}\). Then, there is a subspace \(W\) of dimension \(\leq 2k+2\leq 2Q\) with \(\mathbb{R}^2\times\{0^m\}\subset W\subset \mathbb{R}^{2+m}\) such that \[\label{eqn:soft-subspace-concl} \pi_W^\perp(v) = A_0\log(r) + B_0 + O_{2,1-}(r^{-1/Q+})\tag{28}\] for \(A_0,B_0\in W^\perp\).
Proof. First apply Lemma 7 to deduce \(v = O_{2,1-}(r^{k/Q+}) = O_{2,1-}(r^{1-1/Q+})\). Consider any coordinate direction \(u := \langle v, e\rangle\) for \(e\) a unit vector orthogonal to \(\mathbb{R}^2 \times \{0^m\}\), and suppose that \(u = O_{2,1-}(r^{\ell/Q+})\) for some integer \(\ell\leq k\). Then \(u\) solves \[\label{eqn:soft-subspace-1} g^{ij}(Dv)D_{ij}^2u = 0 \quad \Longrightarrow\quad \Delta u = Dv\star Dv\star D^2u = O_{1-}(r^{-2/Q+\ell/Q-2+}).\tag{29}\] Following the proof of Lemma 6, we can find a particular solution \(u_p = O_{2,1-}(r^{(\ell-2)/Q+})\) to 29 , and hence since \(u-u_p\) is harmonic and admits an expansion as in 21 , we can expand \[u = a_\ell r^{\ell/Q}\cos(\ell\theta/Q) + b_\ell r^{\ell/Q}\sin(\ell\theta/Q) + O_{2,1-}(r^{(\ell-1)/Q+})\] for some \(a_\ell,b_\ell\in\mathbb{R}\). We will repeatedly apply the above principle.
Set \(P_0 := \mathbb{R}^2\times\{0^m\}\). Then expand \[v = A_{k}r^{k/Q}\cos(k\theta/Q) + B_{k} r^{k/Q}\sin(k\theta/Q) + O_{2,1-}(r^{(k-1)/Q+})\] for \(A_{k},B_{k}\in P_0^\perp\). Set \(P_1:= P_0\oplus \mathbb{R}A_{k_0} \oplus \mathbb{R}B_{k_0}\), and note that we have \(\pi_{P_1^\perp}(v) = O_{2,1-}(r^{(k-1)/Q+})\). Now apply the aforestated principle again to expand \[\begin{align} \pi^\perp_{P_1}(v) & = A_{k-1}r^{(k-1)/Q}\cos((k-1)\theta/Q)\\ &+ B_{k-1}r^{(k-1)/Q}\sin((k-1)\theta/Q) + O_{2,1-}(r^{(k-2)/Q+}), \end{align}\] where \(A_{k-1},B_{k-1}\in P_1^\perp\). Now set \(P_2 := P_1 \oplus \mathbb{R}A_{k-1}\oplus \mathbb{R}B_{k-1}\), and repeat unit we either run out of dimensions (i.e. \(W=\mathbb{R}^{2+m}\), which may only happen when \(2k\geq m\)) or we obtain a \((2k+2)\)-dimensional subspace \(W \equiv P_k\) for which 28 holds. This completes the proof. ◻
This section contains our main technical tool, which is the following lemma.
Lemma 9 (Multiplicity \(2\) sheeting). Given \(\epsilon>0\), there exists \(\delta = \delta(m,\epsilon)\in (0,1)\) so that the following holds. Let \(P = \mathbb{R}^2 \times \{0^m\}\), and \(T\) be an area-minimizing integral \(2\)-current in \(C_2(0)\subset\mathbb{R}^{2+m}\) satisfying \[\label{eqn:freq-sheeting-hyp1} (\partial T)\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_2 = 0, \quad E_{T,P}(0,2)\leq\delta^2,\tag{30}\] and, for some \(N_0\in [\epsilon,1-\epsilon]\), \[\label{eqn:freq-sheeting-hyp2} |\Theta_T(0,\rho)-2|\leq\delta, \quad |N_{T,P}(\rho)-N_0|\leq \delta \quad \text{for all }1/4\leq\rho\leq 2.\tag{31}\] Then, up to flipping the orientation of \(T\), we can write \[\label{eqn:freq-sheeting-concl1} T\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(A^2_{1,1/2}\times\mathbb{R}^m) = \llbracket \mathrm{graph}_{2\mathbb{R}^2}(v)\rrbracket,\tag{32}\] for some smooth, minimal, function \(v:2\mathbb{R}^2\cap A_{1,1/2}\to \mathbb{R}^m\) satisfying \[\label{eqn:freq-sheeting-concl2} |E_{T,P}(0,1)^{-1/2}v - (a\sqrt{r}\cos(\theta/2) + b\sqrt{r}\sin(\theta/2))|_{C^2}\leq \epsilon,\tag{33}\] for some \(a,b\in \mathbb{R}^m\) linearly independent. Moreover, we have \(|N_0-1/2|\leq \epsilon\).
Remark 1. With essentially a verbatim proof, a similar sheeting theorem holds for \(T\) near a density \(3\) plane. Precisely, if instead of \(|\Theta_T(0, \rho) - 2| \leq \delta\) one assumes \(|\Theta_T(0, \rho) - 3| \leq \delta\), then either \[T \mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(A^2_{1, 1/2} \times \mathbb{R}^m) = \llbracket \mathrm{graph}_{2\mathbb{R}^2}(v_2) \rrbracket + \llbracket \mathrm{graph}_{\mathbb{R}^2}(v_1) \rrbracket\] or \[T \mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(A^2_{1, 1/2} \times \mathbb{R}^m) = \llbracket \mathrm{graph}_{3\mathbb{R}^2}(v_3) \rrbracket ,\] where, writing \(E = E_{T, P}(0, 1)\), \(E^{-1/2} v_1\) is \(\epsilon\)-close to \(0\), \(E^{-1/2} v_2\) is \(\epsilon\)-close to being \(1/2\)-homogeneous and harmonic, and \(E^{-1/2} v_3\) is \(\epsilon\)-close to being \(k/3\)-homogeneous and harmonic for some \(k \in \{ 1, 2\}\).
Another case we mention is when the plane has general density \(Q \in \mathbb{N}\), but one has a priori the more precise frequency pinching \(|N_0 - p/Q| \leq \epsilon\) for \(p, Q\) coprime and \(\epsilon(Q)\) sufficiently small. In this case, the approximating Dirichlet-minimizer necessarily splits into \(Q\) multiplicity-one sheets, so the arguments of Lemma 9 imply \[T \mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(A^2_{1, 1/2} \times \mathbb{R}^m) = \llbracket \mathrm{graph}_{Q\mathbb{R}^2}(v) \rrbracket\] where \(E^{-1/2} v\) is \(\epsilon\)-close to being \(p/Q\)-homogeneous and harmonic.
We would expect a similar sheeting-type result to hold for general densities \(Q \in \mathbb{N}\) and general frequencies \(N_0 \in [\epsilon, 1-\epsilon]\), but for \(Q \geq 4\) one would likely need some analogue of a (branched) center manifold, and a notion of frequency relative to this center manifold, to deal with sheets which are separated at very different scales. That said, in this more general setting the same proof as Lemma 9 implies that \(T\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(A^2_{1,1/2}\times\mathbb{R}^m)\) decomposes as \(T=T_1+\cdots+T_N\) for some \(N\geq 2\), where each \(T_i\) lies within the \(\epsilon E^{1/2}\)-neighborhood of either the plane \(P\) or some branch of a function modeled on \(z^{p/q}\) for \(p,q\) coprime satisfying \(0<p<q\leq Q\).
We also remark that Krummel–Wickramasekera have informed us of quantitative estimates they have established in the upcoming works [40], [41] generalizing Theorem 3 to area-minimizers near disjoint unions of graphs of \(C^2\) functions. While these refined estimates are not needed in the present work, they could be useful in further investigations, for instance when a (branched) center manifold is needed. We include a brief overview of these estimates in Appendix 9.
Proof. We proceed by contradiction. Suppose for fixed \(\epsilon>0\), there is a sequence \(\delta_i\to 0\) and area-minimizing \(2\)-currents \(T_i\) satisfying 30 and 31 with \(T_i,\delta_i\) in place of \(T,\delta\) respectively, but for which one of the conclusions 32 or 33 fails for all \(i\). Set \(E_i:= E_{T_i,P}(0,1)\), and note that by Corollary 1 we have \[\label{eqn:sheeting-00} \frac{1}{c(m)} E_{T_i, P}(0, 2) \leq E_i \leq 2^4 E_{T_i, P}(0, 2) \quad \forall i \gg 1.\tag{34}\] .
After flipping the orientation of \(T_i\) as necessary, our hypotheses imply that \(T_i\to 2\llbracket \mathbb{R}^2\rrbracket\) in \(C_2\) as both currents and varifolds (cf. Lemma 4). By Theorem 5 (cf. [25]) and 34 , we can find \(\sigma_i\uparrow 1\), Lipschitz functions \(u_i:B^n_{2\sigma_i}\to \mathcal{A}_2(\mathbb{R}^m)\), and closed subsets \(K_i\subset B_{2\sigma_i}^n\) such that \[\label{eqn:sheeting-0} T_i\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(K_i\times \mathbb{R}^m) = \mathbf{v}(u_i) \mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(K_i\times\mathbb{R}^m);\tag{35}\] \[\label{eqn:sheeting-1} |B^2_{2\sigma}\setminus K_i| + \|T_i\|((B^2_{2\sigma}\setminus K_i)\times\mathbb{R}^m) \leq c_\sigma E^{1+\gamma}_i;\tag{36}\] \[\label{eqn:sheeting-2} \sup_{B^2_{2\sigma}}|u_i|\leq c_\sigma E_i^{1/2}, \quad \sup_{B^2_{2\sigma}}|Du_i|\leq c_\sigma E_i^{\gamma}\tag{37}\] for every \(\sigma\in (0,\sigma_i]\), where \(c_\sigma = c_\sigma(m,\sigma)\) and \(\gamma = \gamma(m)\). We note that the first inequality in 37 comes from Lemma 2. We also have for any \(\sigma\in (0,\sigma_i]\), \[\int_{B^2_{2\sigma}}|Du_i|^2 \leq \int_{C_{2\sigma}}|\vec{T}_i-\vec{P}|^2\, d\|T_i\| + c(m,\sigma)E_i^{1+\gamma} \leq c(m,\sigma)E_i.\] Therefore, by [34] (cf. [32]) there is \(w\in W^{1,2}(B^2_2;\mathcal{A}_2(\mathbb{R}^m))\) which is locally Dirichlet-minimizing on \(B_2^2\) for which \(E_i^{-1/2}u_i\to w\) pointwise a.e. on \(B^2_2\), strongly in \(L^2_{\text{loc}}(B^2_2)\), and also \(E_i^{-1/2}|Du_i|\to |Dw|\) in \(L^2_{\text{loc}}(B_2^2)\). Arguing as in [25], we also have the convergence \(N_{T_i,P}(\rho)\to N_w(\rho)\) for every \(\rho\in (1/4,2)\).
From 36 , the choice of normalization, and the strong \(L^2(B^n_1)\) convergence \(E_i^{-1/2}u_i\to w\), we have \(\int_{B_1}|w|^2 = 1\) and so \(w\not\equiv 0\). Our hypotheses also give \(N_w(\rho) \equiv N_0\in (0,1)\) for all \(\rho\in (1/4,2)\). Therefore, \(w|_{A_{2,1/4}}\) is \(N_0\)-homogeneous, and so as \(w\) is Dirichlet-minimizing locally on \(B_2^2\) and in particular has discrete singular set, by analyticity we know that \(w\) must be \(N_0\)-homogeneous on all of \(B_2\). Furthermore, the average \(w_a\) of \(w\) is a single-valued, \(N_0\)-homogeneous harmonic function, but since \(N_0\in (0,1)\) we must have \(w_a\equiv 0\).
Since the singular set of \(w\) is discrete and \(w\) is homogeneous, we must have \(\mathrm{sing}(w)\subseteq\{0\}\) – in fact, we have \(\{x:w(x)=2\llbracket 0\rrbracket\}\subseteq\{0\}\), as this set must also be discrete, and so if this were not the case we would need \(w\equiv 2\llbracket w_a\rrbracket \equiv 2\llbracket 0\rrbracket\), contradicting \(w\not\equiv 0\). Therefore, for every \(x\in B_2^2\setminus \{0\}\), there is a radius \(\rho = \rho(x)>0\) and harmonic functions \(w_1,w_2:B_\rho^2(x)\to \mathbb{R}^m\) so that \(w = \llbracket w_1(y)\rrbracket + \llbracket w_2(y)\rrbracket\) on \(B_\rho^2(x)\), with \(w_1\equiv -w_2\) and \(w_1(y)\neq 0\) for all \(y\in B_\rho^2(x)\).
Let us first argue that we must have \(N_0 = 1/2\). The graph of \(w|_{S^1}\) must be a connected double-cover of \(S^1\) (otherwise, \(w\) would split globally into two single-valued, \(N_0\)-homogeneous harmonic functions, which would necessarily be zero as \(N_0\in (0,1)\), and thus would contradict \(w\not\equiv 0\)). So, we can lift \(w|_{S^1}\) to a single-valued function \(\hat{w}\) defined on \(2S^1\), which extends to \(2\mathbb{R}^2\) as an \(N_0\)-homogeneous harmonic function satisfying, in polar coordinates \(x=r\theta\), \[w(r,\theta) = \llbracket \hat{w}(r,\theta)\rrbracket + \llbracket \hat{w}(r,\theta+2\pi).\rrbracket\] From the Fourier expansion 21 of \(\hat{w}\), we discover that the only possible value for \(N_0\in (0,1)\) is \(N_0 = 1/2\), corresponding to \[\hat{w}(r,\theta) = A\sqrt{r}\sin(\theta/2) + B\sqrt{r}\cos(\theta/2).\]
We now argue that for every \(z\in A^2_{3/2,1/4}\) and \(\epsilon>0\), there is a radius \(\rho = \rho(w,z,\epsilon)>0\) so that, writing \(w|_{B_{2\rho}(z)} = \llbracket w_1\rrbracket + \llbracket w_2\rrbracket\), we can decompose for \(i\gg1\), \[\label{eqn:sheeting-a1} T_i\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_\rho(z) = T_{i,1} + T_{i,2},\quad \text{with}\quad \|T_i\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_{\rho}(z)\| = \|T_{i,1}\| + \|T_{i,2}\|,\tag{38}\] where \(T_{i,j}\) are non-zero, integral \(2\)-currents with zero boundary in \(C_\rho(z)\), which satisfy \[\label{eqn:sheeting-a2} \sup_{y\in C_\rho(z)\cap \mathrm{spt}\|T_{i,j}\|}d(y,\mathrm{graph}_{\mathbb{R}^2}(E_i^{1/2}w_j))\leq \epsilon E_i^{1/2} \quad \text{for }j=1,2.\tag{39}\] From 38 and our original hypothesis that \(T_i\to 2\llbracket \mathbb{R}^n\rrbracket\) in \(C_2\) as both currents and varifolds, it follows that each \(T_{i,j}\) is area-minimizing and \(T_{i,j}\to \llbracket \mathbb{R}^2\rrbracket\) in \(C_\rho(z)\) as \(i\to\infty\), and so from Allard’s regularity theorem \(T_{i,j}\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_{\rho/2}(z)\) is a smooth multiplicity-one graph over \(\mathbb{R}^2\). Using 39 and a covering argument then implies that \(T\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(A^2_{1,1/2}\times\mathbb{R}^m)\) is a smooth, multiplicity-one, graph of a function \(v\) satisfying 33 , which is the desired conclusion of the lemma.
We now prove 38 and 39 . Without loss of generality we can assume \(w_1(z)>w_2(z)\). Set \(h:= w_1(z)-w_2(z)\). For \(\beta = \beta(m,\epsilon)>0\) to be fixed below, ensure that \(\rho = \rho(w,z,\beta)>0\) is chosen sufficiently small so that \[\label{eqn:sheeting-a3} \max\{\mathrm{osc}_{B_{2\rho}(z)}w_1,\mathrm{osc}_{B_{2\rho}(z)}w_2\}\leq\beta h.\tag{40}\] Now write \[F_i := \int_{B^2_{7/4}}\mathcal{G}(E_i^{-1/2}u_i,w)^2\, dx,\] and note that by assumption we have \(F_i\to 0\). From 40 and the definition of \(F_i\) we have \[\begin{align} \int_{C_{7/4}}d(y,\mathrm{graph}_{2\mathbb{R}^2}(E_i^{1/2}w))^2\, d\|T_i\| & \leq c\int_{K_i\times\mathbb{R}^m}\mathcal{G}(u_i,E_i^{1/2}w)^2\, dx + cE_i^{1+\gamma}\\ & \leq c(m)E_i(F_i+E_i^\gamma). \end{align}\] Define then the union of planes \[\boldsymbol{\pi}_i := (E_i^{1/2}w_1(z) + \mathbb{R}^2)\cup (E_i^{1/2}w_2(x) + \mathbb{R}^2).\] Then we get \[\begin{align} \rho^{-n-2}\int_{C_{2\rho}(z)}d(y,\boldsymbol{\pi}_i)\, d\|T_i\| & \leq 2\rho^{-n-2}\int_{C_{2\rho}(z)}d(y,\mathrm{graph}_{2\mathbb{R}^2}(E_i^{1/2}w))^2\, d\|T_i\|\\ &+ c\rho^{-2}E_i\beta^2 h^2\\ & \leq c(m)\rho^{-2}E_i(\rho^{-n}F_i+\rho^{-n}E_i^\gamma+\beta^2h^2). \end{align}\] By the improved \(L^2-L^\infty\) estimate Theorem 3 (applied at scale \(\rho\)) we have that \(\mathrm{spt}\|T_i\|\cap C_\rho(z)\) lies in the \(R_i\)-neighborhood of \(\boldsymbol{\pi}_i\), where \[R_i:= c_1(m)E_i^{1/2}(\rho^{-n/2}F_i^{1/2}+\rho^{-n/2}E_i^{\gamma/2}+\beta h).\] However, noting that the planes in \(\boldsymbol{\pi}_i\) are parallel and at distance \(E_i^{1/2}h\) from each other, we can ensure \(\beta(m)\) is sufficiently small and \(i\gg1\) (depending on \(\rho\)) to get \[R_i<\frac{1}{100}E_i^{1/2}h.\] Therefore, if we define \[T_{i,j}:= T_i \mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}[C_\rho(z)\cap B_{2R_i}(E_i^{1/2}w_j(z)+\mathbb{R}^2)],\] then \((\partial T_{i,j})\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_\rho(z) = 0\) and the decomposition 38 holds. The only thing left to verify is that \(T_{i,j}\neq 0\). Indeed, if not, then we would have (without loss of generality) \(T_i = T_{i,1}\) for infinitely many \(i\), and then we would get a contradiction since for such \(i\) we would have \[\frac{1}{2}E_ih^2 \leq \frac{1}{\|T_i\|(C_\rho(z))}\int_{C_\rho(z)}d(y,\boldsymbol{\pi}_i)^2\, d\|T_i\| \leq R_i^2 < \frac{1}{100}E_ih^2\] (indeed, in this case one would need the graph of \(w\) in \(C_\rho(z)\) to lie within the \(\frac{1}{100}h\)-neighborhood of the plane \(w_1(z)+\mathbb{R}^2\) which is clearly false). So, we must have \(T_{i,j}\neq 0\) for each \(i,j\), which proves 38 . The estimate 39 then follows by ensuring \(\beta = \beta(m,\epsilon)\) is sufficiently small. We also note that one can take \(\rho = \rho(w,\beta)\) in this argument, i.e. \(\rho\) is independent of the point \(z\in A^2_{1,1/2}\), because one has a uniform lower bound on \(w_1-w_2\), as well as an upper bound on \(|Dw_i|\), on \(A^2_{1,1/2}\). ◻
We now use Lemma 9 to prove an Allard-type sheeting theorem at infinity for area-minimizers with density \(2\) at infinity.
Theorem 8 (Regularity near infinity). Suppose \(T\) is a non-planar, area-minimizing \(2\)-current in \(\mathbb{R}^{2+m}\) with \(\partial T = 0\) and \(\Theta_T(\infty)=2\). Then, there is \(\alpha = \alpha(m)\in (0,1)\) such that, after a possible rotation and dilation in \(\mathbb{R}^{2+m}\), one of the following cases occurs:
\(T\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(\mathbb{R}^{2+m}\setminus \overline{C_1}) = \llbracket \mathrm{graph}_{2\mathbb{R}^2}(v)\setminus \overline{C}_1\rrbracket\), for \(v:2\mathbb{R}^2\setminus \overline{B_1}\to \mathbb{R}^m\) a smooth function satisfying \(v = O(r^{1-\alpha+})\);
\(T\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(\mathbb{R}^{2+m}\setminus \overline{B_1}) = (\llbracket \mathrm{graph}_{P_1}(u_1)\rrbracket + \llbracket \mathrm{graph}_{P_2}(u_2)\rrbracket)\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(\mathbb{R}^{2+m}\setminus \overline{B_1})\), for \(P_1,P_2\) transverse \(2\)-planes and \(u_i:P_i\to P_i^\perp\) smooth functions satisfying \(u_i = O(r^{1-\alpha+})\).
Proof. Let \(\mathbf{C}= \llbracket P_1\rrbracket + \llbracket P_2\rrbracket\) be the tangent cone of \(T\) at infinity; we know by Theorem 6 that \(\mathbf{C}\) is unique and, there exists \(\alpha = \alpha(m)\in (0,1)\) such that for any \(\eta>0\) and \(\alpha^\prime<\alpha\) we have \[\label{eqn:structure-1} d_H(\mathrm{spt}\|T\|\cap B_R,\mathrm{spt}\|\mathbf{C}\|\cap B_R)\leq R^{1-\alpha} \leq \eta R^{1-\alpha^\prime} \quad \text{for all }R\gg 1.\tag{41}\] We also know that either \(P_1 = P_2\) or \(P_1\cap P_2=\{0\}\) (as \(\mathbf{C}\) is area-minimizing).
If \(P_1\neq P_2\), then \(P_1\cap P_2=\{0\}\), and so conclusion (2) follows from Allard’s regularity theorem and 41 . Hence, after rotating we may suppose \(P_1 = P_2 = P = \mathbb{R}^2 \times \{0^m\}\). From Corollary 1 and 41 , provided \(\eta = \eta(m,\alpha^\prime)\) is sufficiently small (which we may assume), the quantity \[N_{T,P}(\rho)\exp(c \eta^{2\gamma}\rho^{-2\alpha^\prime\gamma} + c'\eta^{2\gamma-2\gamma_1}(\rho^{-2} D_{T,P}(\rho))^{\gamma_1})\] is non-decreasing for \(\rho\gg 1\) (for suitable constants \(c, c'\)), and hence has a limit \(M\in [0,\infty]\). Since this quantity is \(>0\) (it is well-defined as \(T\) is not planar) we know \(M>0\). Moreover, since by Lemma 3 and 41 we have \[\rho^{-2}D_{T,P}(\rho) \leq \frac{1}{2}\rho^{-n}\int_{C_\rho}\|\pi_T-\pi_P\|^2\, d\|T\| \leq cE_{T,P}(0,2\rho) \to 0\] and get that \(M = \lim_{\rho\to\infty} N_{T,P}(\rho)\) also.
We now claim that \(M\leq1-\alpha\). To see this, we argue by contradiction: suppose that \(M>1-\alpha\). If \(M<\infty\), set \(M^\prime := (M+1-\alpha)/2>1-\alpha\), otherwise if \(M=\infty\), then set \(M^\prime:=1\). Then, choosing \(\epsilon= \epsilon(m,M,M^\prime,\alpha^\prime)\) sufficiently small and \(\tau = \tau(\epsilon,T)\) sufficiently large, from Corollary 1 we have, if \(M<\infty\) \[\label{eqn:structure-2} H_{T,P}(\rho) \geq (\rho/\tau)^{2(M-\epsilon)-c\epsilon^{2\gamma}}H_{T,P}(\tau) \geq c(\tau,\epsilon,M,T)\rho^{2M^\prime} \quad \text{for all }\rho>\tau.\tag{42}\] for some constant \(c(\tau,\epsilon,M,T)>0\) (of course we know that \(H_{T,P}(\tau)>0\) as \(T\) is assumed to be non-planar); if instead \(M=\infty\), the same argument works just replacing \(M-\epsilon\) in the first inequality by \(2\), say. However, on the other hand, from 41 we have, for all \(\rho\gg 1\), \[\begin{align} H_{T,P}(\rho) & \leq 2\rho^{1-2}\int_{C_\rho\setminus C_{\rho/2}}\left(\sup_{x\in\mathrm{spt}\|T\|\cap C_{2\rho}}d_P(x)^2\right)|\nabla r|^2\frac{1}{r}\, d\|T\|(x)\\ & \leq c\rho^{-1}\cdot \rho^{2-2\alpha}\cdot\frac{1}{\rho}\cdot\|T\|(C_\rho)\\ & \leq c\rho^{2-2\alpha}. \end{align}\] However combining this inequality with 42 gives \(c\rho^{2M^\prime}\leq \rho^{2-2\alpha}\) for all \(\rho\gg 1\), which is a contradiction since \(M^\prime>1-\alpha\). Hence, this shows that we must have \(M\leq 1-\alpha\).
Hence, we have now shown that there is an \(\epsilon'>0\) for which \(M\in [\epsilon',1-\epsilon']\). Since \(N_{T,P}(\rho)\to M\), \(\Theta_T(0,\rho)\to 2\), and \(E_{T,P}(0,\rho)\to 0\) as \(\rho\to\infty\), we can therefore apply Lemma 9 on large cylinders to deduce that \(T\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(\mathbb{R}^{2+m}\setminus \overline{C_R}) = \llbracket \mathrm{graph}_{2\mathbb{R}^2}(v)\setminus \overline{C_R}\rrbracket\) for some large \(R>1\) and some smooth \(v:2\mathbb{R}^2\setminus \overline{B_1}\to \mathbb{R}^m\). The asymptotics \(v = O(r^{1-\alpha+})\) then follow from 41 . ◻
In this section we show that any entire, indecomposable, area-minimizing integral \(2\)-current \(T\) with \(\Theta_T(\infty)<\infty\) and is regular at infinity is contained within an affine subspace of dimension \(\leq 2\Theta_T(\infty)\). Here, by “indecomposable”, we mean we cannot write \(T = T_1+T_2\), \(\|T\| = \|T_1\| + \|T_2\|\) with \(T_1,T_2\) non-zero (necessarily area-minimizing) integral \(2\)-currents without boundary.
Our first step is the following “Liouville”-type theorem for two-dimensional stationary integral varifolds which are contained within a slab-region. This result is essentially a classical fact about bounded harmonic functions, but for the reader’s convenience we give the proof in the context of stationary varifolds.
Lemma 10. Let \(V\) be a stationary \(2\)-varifold in \(\mathbb{R}^{2+m}\) with \(\Theta_V(\infty)<\infty\). If for some unit vector \(e\) we have \(\sup_{\mathrm{spt}\|V\|}|\langle e,x\rangle|<\infty\), then \(\nabla_V\langle e,x\rangle =0\) at \(\|V\|\)-a.e. \(x\).
Remark 1. Lemma 10 is false without the assumption \(\Theta_V(\infty)<\infty\) (see [42]). It is also false of \(V\) has dimension \(\geq 3\), as the standard catenoid in \(\mathbb{R}^{n+1}\) for \(n\geq 3\) is contained within two parallel planes. In the present lemma, one can weaken the growth rate on \(|\langle e,x\rangle|\) to sub-logarithmic.
Proof. After a rotation there is no loss in generality in assuming that \(e=e_1\), so that \(\langle e,x\rangle = x_1\) is the first coordinate. Take \(\phi\in C^1_c(\mathbb{R})\) and choose the vector field \(X = \phi^2(|x|)x_1e_1\) in the first variation formula for \(V\). Using that \(\pi_{T_xV}(e_1) = \nabla x_1|_x\), we obtain \[\int\phi^2|\nabla x_1|^2\, d\|V\| = -\int2\phi\phi^\prime\big\langle \pi_{T_xV}(x/|x|), e_1\big\rangle x_1\, d\|V\|\] which therefore gives, by the Cauchy–Schwarz inequality, \[\int\phi^2|\nabla x_1|^2\, d\|V\| \leq 4\int(\phi^\prime)^2x_1^2\, d\|V\|.\] Let \(\phi(r)\) be a (smooth approximation of) a log-cutoff function at radius \(R\), i.e. \[\phi(r) = \begin{cases} 1 & \text{if }r\leq R,\\ 1-\frac{\log(r)-\log(R)}{\log(R^2)-\log(R)} & \text{if }R<r<R^2,\\ 0 & \text{if }R^2\leq r. \end{cases}\] Then we get \[\begin{align} \int_{B_R}|\nabla x_1|^2\, d\|V\| & \leq 4\int_{B_{R^2}\setminus B_R}\frac{1}{r^2(\log(R))^2}x_1^2\, d\|V\|\\ &\leq \frac{4\sup_{\mathrm{spt}\|V\|}x_1^2}{(\log(R))^2}\int_{B_{R^2}\setminus B_R}\frac{1}{r^2}\, d\|V\|\\ & \leq \frac{32\pi\Theta_V(\infty)\cdot\sup_{\mathrm{spt}\|V\|}|x_1|^2}{\log(R)} \end{align}\] where the last inequality follows from a standard dyadic-scale trick, since \(\|V\|(B_r)\leq \Theta_V(\infty)\pi r^2\) for all \(r>0\). Taking \(R\to\infty\) then gives \(\nabla x_1 = 0\) \(\|V\|\)-a.e., which completes the proof. ◻
The following is the main result of this section.
Theorem 9 (Subspace rigidity). Suppose \(T^2\) is an indecomposable2 area-minimizing integral \(2\)-current in \(\mathbb{R}^{2+m}\) with \(\partial T =0\) and which is regular at infinity in the following sense: there are \(\alpha>0\), \(Q_1,\dotsc,Q_\ell\in \mathbb{N}\), smooth functions \(\{v_i:Q_i\mathbb{R}^2\to \mathbb{R}^m\}_{i=1}^\ell\), and rigid motions \(O_1,\dotsc,O_\ell\in O(2+m)\), so that \[\label{eqn:subspace-hyp} T\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(\mathbb{R}^{2+m}\setminus \overline{B}_1) = \sum_{i=1}^\ell (O_i)_\#\llbracket \mathrm{graph}_{Q_i\mathbb{R}^2}(v_i)\rrbracket \mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(\mathbb{R}^{2+m}\setminus \overline{B}_1) \quad \text{with }v_i = O(r^{1-\alpha+}).\tag{43}\] Then, there is a subspace \(W\subset\mathbb{R}^{2+m}\) of dimension \(\leq 2\Theta_T(\infty)\) and \(p\in \mathbb{R}^{2+m}\) so that \(\mathrm{spt}\|T\| \subset p+W\).
Proof. Set \(P_i := O_i(\mathbb{R}^2)\), oriented correctly so that the tangent cone of \(T\) at infinity is \(\mathbf{C}= \sum^\ell_{i=1}Q_i\llbracket P_i\rrbracket\). We will abuse notation slightly and think of \(v_i(r,\theta)\) as functions on \(P_i\), i.e. so that \(v_i:Q_iP_i\to P_i^\perp\) and \[T_i\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(\mathbb{R}^{2+m}\setminus \overline{B_1}) = \sum^\ell_{i=1}\llbracket \mathrm{graph}_{Q_iP_i}(v_i)\rrbracket\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(\mathbb{R}^{2+m}\setminus \overline{B_1}).\] Write \(Q := \max_i Q_i\) and define \(\eta:= \min_{P_i \neq P_j} d_H(P_i\cap B_1,P_j\cap B_1)\).
Noting that \(\Theta_T(\infty) = \sum^\ell_{i=1}Q_i\), we can apply Lemma 8 to each function \(v_i\) to obtain a subspace \(W\) of dimension \(\leq 2\Theta_T(\infty)\) so that, for every \(i=1,\dotsc,\ell\), \[\label{eqn:subspace-0465} \pi^\perp_W(v_i) = A_{i}^\perp\log(r) + B_{i}^\perp + O_{2,1-}(r^{-1/Q+})\tag{44}\] for some \(A_{i}^\perp,B_{i}^\perp\in W^\perp\).
We claim that \(A_{i}^\perp=0\) for each \(i\). To prove this, we will suppose not and find a contradiction to the area-minimizing property of \(T\) by constructing a suitable competitor. Define \(v_{i,0} = v_i - A_i^\perp \log(r)\). Then from Lemma 7 and 44 we get \[\begin{align} \langle D_k v_i , D_l v_i\rangle &= \langle D_k v_{i,0}, D_l v_{i, 0}\rangle + \frac{|A_i^\perp|^2}{r^2} (D_k r)(D_l r) + O(r^{-1/Q-2+}) \tag{45} \\ &= O(r^{-2/Q+}) \tag{46} \end{align}\] for \(k, l \in \{ 1, 2 \}\).
Take \(1\ll \rho\ll R\) both large, to be fixed later. Ensure \(\rho\) is sufficiently large so that, on \(\{r\geq\rho\}\), \[\begin{align} \max\{|Dv_i| , |Dv_{i,0}| \} \leq 1/100 \quad \text{ and } \quad \max\{ |v_i|, |v_{i,0}|\} \leq \eta r /100.\label{eqn:subspace-1} \end{align}\tag{47}\] Let \(\phi:\mathbb{R}\to\mathbb{R}\) be a smooth, increasing function obeying \(\phi\equiv 0\) on \((-\infty,\rho]\), \(\phi\equiv 1\) on \([2\rho,\infty)\), and \(\phi^\prime\leq 2/\rho\) on \(\mathbb{R}\). Define \(w_i:Q_iP_i\to P_i^\perp\) by \[w_i(r,\theta):= \begin{cases} v_{i,0}(r,\theta)\phi(r) + A_{i}^\perp\log(R) & \text{if }r\geq \rho,\\ A_{i}^\perp\log(R) & \text{if }r<\rho. \end{cases}\] We have \[\label{eqn:subspace-2} w_i = v_i \quad \text{on }P_i\cap \partial B_R,\tag{48}\] and \[D_kw_i = D_kv_{i,0} \quad \text{for }r\geq 2\rho, \quad\text{and}\quad |Dw_i|^2 \leq c(v_i) \quad \text{for }r\in [\rho,2\rho], \label{eqn:subspace-2461}\tag{49}\] where \(c(v_i)\) denotes a constant depending on \(v_i\) but independent of \(\rho,R\).
Note that \(w_i\) coincides with a single-valued constant (with multiplicity \(Q_i\)) for \(r<\rho\), so the currents \(\llbracket \mathrm{graph}_{Q_iP_i}(w_i)\rrbracket\) are entirely smooth and without boundary. Moreover, by 47 and 48 , we have \[\partial(T\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}\cap^\ell_{i=1}C_R(0,P_i)) = \partial\left(\sum^\ell_{i=1}\llbracket \mathrm{graph}_{Q_iP_i}(w_i)\cap C_R(0,P_i) \rrbracket\right)\] and hence, by minimality of \(T\), we must have \[\label{eqn:subspace-3} \|T\|(\cap^\ell_{i=1}C_R(0,P_i)) \leq \sum^\ell_{i=1}\|\mathrm{graph}_{Q_iP_i}(w_i)\|(C_R(0,P_i)).\tag{50}\]
On the other hand, using 45 , 46 with the relation \[\sqrt{\det(\mathrm{Id} + G + H)} = \sqrt{\det(\mathrm{Id}+G)} (1+\mathrm{tr}(H)/2+O(|G||H|) + O(|H|^2))\] for \(2\times 2\) matrices \(G, H\) satisfying \(\max \{|G|, |H|\} \leq 1/100\), and then using 49 , we can compute: \[\begin{align} \|T\|&(\cap ^\ell_{i=1}C_R(0,P_i))\\ & \geq \sum^\ell_{i=1}\int_{Q_iP_i\cap (B_R\setminus B_\rho)} Jv_i \, dx\\ & \geq\sum^\ell_{i=1} \int_{Q_iP_i\cap (B_R\setminus B_\rho)}J v_{i,0} + \frac{|A_{i}^\perp|^2}{2r^2} + O(r^{-1/Q-2+})\, dx\\ & \geq \sum^\ell_{i=1} \left( \int_{Q_iP_i\cap (B_R\setminus B_\rho)} J v_{i,0}\, dx + \pi Q_i\log(R/\rho)|A_{i}^\perp|^2 - c(v_i) \right) \\ & \geq\sum^\ell_{i=1} \left( \int_{Q_iP_i\cap (B_R\setminus B_\rho)} Jw_i \, dx + \pi Q_i|A_{i}^\perp|^2\log(R/\rho) - c(v_i)\rho^2 \right)\\ & \geq \sum^{\ell}_{i=1} \|\mathrm{graph}_{Q_iP_i}(w_i)\|(C_R(0,P_i)) + \sum_{i=1}^\ell \Big( \pi Q_i| A^\perp_{i}|^2\log(R/\rho) - c(v_i) \rho^2 \Big) \\ & > \sum^\ell_{i=1}\|\mathrm{graph}_{Q_iP_i}(w_i)\|(C_R(0,P_i)) \end{align}\] where the last inequality if strict some \(A^\perp_i \neq 0\) and \(R\) is chosen sufficiently large (depending on the \(v_i\) and \(\rho\)). This contradicts 50 , and hence proves our claim.
Now, since each \(A_{i}^\perp=0\), from the expansion 44 , we see that we must have \(\sup_{x\in \mathrm{spt}\|T\|}|\langle x,e\rangle|<\infty\) for all \(e\in W^\perp\). Hence, from Lemma 10, we get that \(\langle x,e\rangle\) is locally constant on \(\mathrm{spt}\|T\|\), and hence is constant by indecomposability of \(T\). It therefore follows that \(\mathrm{spt}\|T\|\subset p+W\) for some \(p\in \mathbb{R}^{2+m}\). ◻
We can now combine all we have done to prove Theorem 1.
Proof of Theorem 1. If \(\Theta_T(p) \geq 2\) at any \(p\) then since \(\Theta_T(\infty) = 2\) the monotonicity formula implies \(T = \llbracket p + P_1 \rrbracket + \llbracket p + P_2 \rrbracket\) for either \(P_1 = P_2\) or (by the area-minimizing property) \(P_1 \cap P_2 = \{0\}\). In the latter case, \(P_1, P_2\) must be jointly holomorphic for some orthogonal complex structure by [17].
By Allard, let us therefore assume \(T\) is entirely regular and multiplicity-one. If \(T\) were decomposable, that is \(T = T_1 + T_2\) and \(\|T\| = \|T_1\| + \|T_2\|\) for \(T_i\) non-zero and \(\partial T_i = 0\), then each \(T_i\) would be area-minimizing and hence we must have \(\Theta_{T_i}(\infty) = 1\). So each \(T_i\) is a multiplicity-one plane, and we deduce \(T = \llbracket p_1 + P_1 \rrbracket + \llbracket p_2 + P_2 \rrbracket\), where as before either \(P_1 = P_2\) or \(P_1 \cap P_2 = \{0\}\) are jointly holomorphic up to a real rigid motion.
We may assume \(T\) is indecomposable. By Theorem 8 we know \(T\) is regular at infinity, and so we can apply Theorem 9 to deduce \(T\) is supported in some affine copy of \(\mathbb{R}^4 \equiv \mathbb{C}^2\). It then follows from [19] that \(\mathrm{spt}\|T\|\) is algebraic and, up to a further rigid motion, the zero set of a degree \(2\) holomorphic polynomial, which completes the proof.
For the sake of completeness we outline the argument, which simplifies in this special case. From Micallef’s result [1] we know \(T\) is holomorphic up to rigid motion, and hence by the [24]’s uniqueness of tangent cone at infinity we can extent \(\mathrm{spt}\|T\|\) to a closed, complex subvariety of \(\mathbb{CP}^2\). Chow’s theorem implies \(\mathrm{spt}\|T\|\) must be algebraic, and having density \(2\) at infinity forces the degree of the subsequence (irreducible) polynomial to be \(2\). ◻
We also justify the claim in the introduction that if \(T^2\subset\mathbb{R}^n\) is a complete area-minimizer asymptotic to \(Q\llbracket P\rrbracket\) (so in particular \(\Theta_T(\infty)=Q\)) and \(N_{T,P}(\infty) = 1/Q\), then \(T\) must be contained in some affine copy of \(\mathbb{R}^4\), and thus is algebraic by [19]. Indeed, in this situation one can prove that \(T\) is regular at infinity by Remark 1, at which point the condition \(N_T(\infty)=1/Q\) forces the asymptotic expansion from Lemma 7 to become \[v(r,\theta) = ar^{1/Q}\cos(\theta/Q)+br^{1/Q}\sin(\theta/Q) + c\log(r)+d + O(r^{-1/Q}).\] The proof from Theorem 9 then implies that \(T\) is contained in an affine copy of \(W:= \mathbb{R}^2\oplus \mathbb{R}a\oplus \mathbb{R}b \cong \mathbb{R}^4\). 0◻
We prove the following \(L^2\) estimate which was used in the proof of Theorem 6.
Lemma 11. Fix \(0<\sigma<\tau\). Let \(V\) be a stationary \(n\)-varifold in \(A_{\tau,\sigma/2}\subset\mathbb{R}^{n+m}\), and let \(f:\mathbb{R}^{n+m}\setminus\{0\}\to \mathbb{R}\) be a \(1\)-homogeneous \(C^1\) function admitting the bound \(\|f\|_{C^1(S^{n+m-1})}\leq M\). Set \(\Lambda:= \sup_{\sigma\leq\rho\leq\tau}\rho^{-n}\|V\|(A_{\rho,\rho/2})\). Then we have the inequality \[\begin{align} &\left|\tau^{-n-2}\int_{A_{\tau,\tau/2}}f(x)^2\, d\|V\| - \sigma^{-n-2}\int_{A_{\sigma,\sigma/2}}f(x)^2\, d\|V\|\right|\\ &\leq \int_{A_{\tau,\tau/2}\cup A_{\sigma,\sigma/2}}\frac{|\pi^\perp_V(x)|^2}{|x|^{n+2}}\frac{f(x)^2}{|x|^2}\, d\|V\|\\ & + 4M^2\Lambda^{1/2}\log(\tau/\sigma)\left(\int_{A_{\tau,\sigma/2}}\frac{|\pi_V^\perp(x)|^2}{|x|^{n+2}}\, d\|V\|\right)^{1/2}. \end{align}\]
Proof. Take \(\zeta:\mathbb{R}\to\mathbb{R}\) a non-negative, smooth function compactly supported in \([1/2,1]\). Fix also \(\rho\in [\sigma,\tau]\). Consider the vector field \[X(x) := \zeta(|x|/\rho)f(x)^2 x.\] Note that since \(f(x)\) is \(1\)-homogeneous we have \(\langle x,D(f^2)\rangle=2f^2\). Let us abbreviate \(x^\top \equiv \pi^{\top}_V(x)\) and \(x^\perp\equiv \pi_V^{\perp}(x)\). We then compute \[\begin{align} \textrm{div}_V(X) & = \zeta^\prime(|x|/\rho)\frac{|x^\top|^2}{|x|\rho}f(x)^2 + \zeta(|x|/\rho)\langle x,\nabla_V(f^2)\rangle + n\zeta(|x|/\rho)f(x)^2\\ & = \zeta^\prime f^2\frac{|x|}{\rho} - \zeta^\prime f^2\frac{|x^\perp|^2}{|x|\rho} + (n+2)\zeta f^2 - \zeta\langle x,\nabla_V^\perp(f^2)\rangle, \end{align}\] and therefore, by stationarity of \(V\), \[\begin{align} \frac{d}{d\rho}\left(\rho^{-n-2}\int\zeta(|x|/\rho)f^2\, d\|V\|\right) & = -\rho^{-n-3}\int (n+2)\zeta f^2 + \zeta^\prime f^2\frac{|x|}{\rho}\, d\|V\|\\ & = -\rho^{-n-3}\int \zeta^\prime f^2\frac{|x^\perp|^2}{|x|\rho} + \zeta\langle x,\nabla^\perp_V (f^2)\rangle\, d\|V\|. \end{align}\] Integrating this over \(\rho\in [\sigma,\tau]\), and then taking \(\zeta\to \mathbf{1}_{[1/2,1]}\), we get \[\begin{align} &\tau^{-n-2}\int_{A_{\tau,\tau/2}}f^2\, d\|V\| - \sigma^{-n-2}\int_{A_{\sigma,\sigma/2}}f^2\, d\|V\|\\ &= \int_{A_{\tau,\sigma}}\frac{|x^\perp|^2f^2}{|x|^{n+4}}\, d\|V\| - \int_{A_{\tau/2,\sigma/2}}\frac{|x^\perp|^2f^2}{|x|^{n+4}}\, d\|V\|\\ &- \int^\tau_\sigma\rho^{-n-3}\int_{A_{\rho,\rho/2}}\langle x,\nabla^\perp_V(f^2)\rangle\, d\|V\|d\rho. \end{align}\]
From the \(1\)-homogeneity of \(f\), we have \[|\langle x,\nabla^\perp_V(f^2)\rangle| = |\langle\nabla^\perp_V(x),D(f^2)\rangle| \leq 4|x^\perp||x|M^2,\] and so we can bound the last term like \[\begin{align} &\left|\int^\tau_\sigma\rho^{-n-3}\int_{A_{\rho,\rho/2}}\langle x,\nabla^\perp_V(f^2)\rangle\, d\|V\| d\rho\right|\\ & \leq 4M^2\int^\tau_{\sigma}\rho^{-n-3}\left(\int_{A_{\rho,\rho/2}}|x|^2\, d\|V\|\right)^{1/2}\left(\int_{A_{\rho,\rho/2}}|x^\perp|^2\, d\|V\|\right)^{1/2}\, d\rho\\ & \leq 4M^2\Lambda^{1/2}\int^\tau_\sigma\rho^{-1}\left(\rho^{-n-2}\int_{A_{\rho,\rho/2}}|x^\perp|^2\, d\|V\|\right)^{1/2}\, d\rho\\ & \leq 4M^2\Lambda^{1/2}\int^\tau_\sigma\rho^{-1}\left(\int_{A_{\rho,\rho/2}}\frac{|x^\perp|^2}{|x|^{n+2}}\, d\|V\|\right)^{1/2}\, d\rho\\ & \leq 4M^2\Lambda^{1/2}\log(\tau/\sigma)\left(\int_{A_{\tau,\sigma/2}}\frac{|x^\perp|^2}{|x|^{n+2}}\, d\|V\|\right)^{1/2}. \end{align}\] This completes the proof. ◻
In this section we prove almost-monotonicity of Krummel–Wickramasekera’s planar frequency function, in the form stated in Theorem 7. Our argument is a refinement of that in [25]. The main technical differences are how the graphical scales \(\sigma(\xi)\) are chosen, and how the error of \(D^\prime\) is handled, which together allow us to show the “scale invariant” differential inequality 19 for general small excess \(E\). Throughout this section we fix \(P := \mathbb{R}^n\times\{0^m\}\). Also, for a \(q\)-valued function \(w = \sum_{i=1}^q\llbracket w_i\rrbracket:B_1^n\to \mathcal{A}_q(\mathbb{R}^m)\) and \(p\in \mathbb{R}^m\), we define \(w\ominus p := \sum^q_{i=1}\llbracket w_i-p\rrbracket\).
Lemma 12. Given \(a>0\), there is \(b = b(n,m,q,a)>0\) such that the following holds. Let \(w\in W^{1,2}(B^n_1;\mathcal{A}_q(\mathbb{R}^m))\) be a Dirichlet-minimizing function. Then: \[\begin{align} & \|w\|_{L^2(B_{1/2})} \geq a\|w\|_{L^2(B_1)}\nonumber\\ &\Longrightarrow\quad \|w\|_{L^2(B_{1/20}(\xi))}\geq b\|w\|_{L^2(B_1)} \quad \text{for all }\xi\in B_{1/5};\label{eqn:recenter-concl1} \end{align}\tag{51}\] and \[\begin{align} & \|Dw\|_{L^2(B_{1/2})} \geq a\|Dw\|_{L^2(B_1)}\nonumber\\ &\Longrightarrow\quad \|Dw\|_{L^2(B_{1/20}(\xi))}\geq b\|Dw\|_{L^2(B_1)}\quad \text{for all }\xi\in B_{1/5}.\label{eqn:recenter-concl2} \end{align}\tag{52}\]
Proof. Implication 51 is a straightforward contradiction argument, as in [25]. For implication 52 , we can again argue by contradiction as follows. If the claim were false, then for some fixed \(a > 0\) we could find sequences of numbers \(b_j \to 0\), points \(\xi_j \to \xi \in \overline{B_{1/5}^n}\) and Dirichlet-minimizers \(w_j : B_1^n \to \mathcal{A}_Q(\mathbb{R}^m)\) so that \[\begin{align} &\|Dw_j\|_{L^2(B_1)} = 1 \quad \text{and}\quad \|D w_j\|_{L^2(B_{1/2})} \geq a \tag{53} \\ &\quad \text{yet} \quad \|Dw_j\|_{L^2(B_{1/40}(\xi))} \leq \|D w_j\|_{L^2(B_{1/20}(\xi_i))} \leq b_j \to 0. \tag{54} \end{align}\]
We may then apply the compactness theorem for Dirichlet-minimizers with unit Dirichlet-energy (see [32]) to deduce that, up to passing to another subsequence, we can find \(J\in\{1,\dotsc,Q\}\), positive integers \(k_1,\dotsc,k_J\in\{1,\dotsc,Q\}\) with \(k_1+\cdots+k_J = Q\), points \(p_{j}^{(1)},\dotsc,p_j^{(J)}\in \mathbb{R}^m\), and Dirichlet-minimizers \(g^{(1)},\dotsc,g^{(J)} :B^n_1\to \mathcal{A}_{k_i}(\mathbb{R}^m)\) such that we have \[w_j = \sum^J_{i=1}w_j^{(i)} \ominus p_j^{(i)}\] where \(w_j^{(i)}\) are Dirichlet-minimizers such that for every \(i = 1, \ldots, J\), we have \(w_j^{(i)} \to g^{(i)}\) locally uniformly in \(B_1^n\) and \(|Dw_j^{(i)}| \to |Dg^{(i)}|\) in \(L^2_{\text{loc}}(B_1)\). In particular, \[\sum^J_{i=1}\int_{B^n_\sigma(y)}|Dg^{(i)}|^2 = \lim_{j\to\infty}\int_{B^n_{\sigma}(y)}|Dw_j|^2\] for each ball \(B^n_\sigma(y)\subset\subset B^n_1\). Our assumptions 53 , 54 then give \[\sum^J_{i=1}\int_{B_{1/2}}|Dg^{(i)}|^2 \geq a\quad \text{yet}\quad \sum^J_{i=1}\int_{B_{1/40}(\xi)}|Dg^{(i)}|^2=0.\] However, unique continuation (cf. [32]) then implies that we must have \(\|Dg^{(i)}\|_{L^2(B_1)}=0\) for all \(i\), giving the desired contradiction. ◻
Lemma 13. Let \(q\in \mathbb{N}\), \(a>0\), and let \(T\) be an area-minimizing integral \(n\)-current in \(C_8(0)\) satisfying \[(\partial T)\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_\infty(0) = 0, \quad \pi_{P\#}T = q\llbracket B^n_\infty(0)\rrbracket,\quad E_{T,P}(0,8)\leq\epsilon^2,\] \[\int_{C_{1/2}(0)}d_P^2\, d\|T\| \geq a^2\int_{C_8(0)}d_P^2\, d\|T\|.\] Then, provided \(\epsilon= \epsilon(n,m,q,a)\in (0,1)\) is sufficiently small, there is \(b = b(n,m,q,a)>0\) for which \[\int_{C_{1/20}(\xi)}d_P^2\, d\|T\| \geq b^2\int_{C_8(0)}d_P^2\, d\|T\| \quad \text{for all }\xi\in B^n_{1/5}(0).\]
Proof. By Lemma 4 and Lemma 2 we have \[\sup_{x\in \mathrm{spt}\|T\|\cap C_7(0)} d_P(x)\leq c\epsilon\quad \text{and}\quad \mathcal{E}_{T,P}(0,6)\leq c\epsilon^2.\] From these, ensuring \(\epsilon= \epsilon(n,m,q)\) is sufficiently small, we can apply Almgren’s Lipschitz approximation (Theorem 5) to approximate \(T\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_1\) by a \(q\)-valued Lipschitz graph. The proof then proceeds verbatim as in [25], with the help of Lemma 12. ◻
Lemma 14. Let \(T\) be an area-minimizing integral \(n\)-current in \(C_{2\rho}(0)\). Suppose that \[\label{eqn:H-error-hyp1} (\partial T)\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_{2\rho}(0) = 0, \quad \Theta_T(0,\rho/2)\geq 1/2, \quad \Theta_T(0,2\rho) \leq \theta,\tag{55}\] \[E_{T,P}(0,2\rho) \leq E \leq \delta^2.\] Then, provided \(\delta = \delta(n,m,\theta)\in (0,1)\) is sufficiently small we have \[\label{eqn:H-error-concl1} -\rho^{1-n}\int d_P^2\|\pi_T-\pi_P\|^2\frac{1}{r}\phi^\prime(r,\rho)\, d\|T\| \leq cE^\gamma H_{T,P}(\rho),\tag{56}\] and therefore \[\label{eqn:H-error-concl2} \left|H^\prime_{T,P}(\rho)+2\rho^{-n}\int\langle x,\pi_{P^\perp}(\nabla^Tr)\rangle\phi^\prime(r/\rho)\, d\|T\|\right| \leq cE^\gamma \rho^{-1}H_{T,P}(\rho),\tag{57}\] and \[\label{eqn:H-error-concl3} H_{T,P}(\rho) \leq 2\rho^{1-n}\int_{C_\rho\setminus C_{\rho/2}}d_P^2\frac{1}{r}\, d\|T\| \leq (1+cE^{\gamma})H_{T,P}(\rho).\tag{58}\] Here, \(c = c(n,m,\theta)>0\) and \(\gamma = \gamma(n,m,\theta)>0\).
Proof. We follow [25]. From Lemma 4 and Lemma 2 we have \[\label{eqn:H-error-0466} \mathcal{E}_{T,P}(0,\rho)\leq c(n,m)E \quad \text{and} \quad \sup_{x\in \mathrm{spt}\|T\|\cap C_\rho(0)}d_P(x) \leq c(n,m)E^{1/2}\rho.\tag{59}\] From Lemma 5, provided \(\delta = \delta(n,m,\theta)\) is sufficiently small, we have \[\label{eqn:H-error-0465} \pi_{P\#}T = q\llbracket B^n_{2\rho}(0)\rrbracket\tag{60}\] for some integer \(|q|\leq\theta\). If \(q=0\) then by 59 and Lemma 5 we must have \(T\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_\rho(0)=0\), but this contradicts our lower density assumption from 55 . So, after possibly flipping the orientation of \(T\) as necessary, we may assume \(1\leq q\leq\theta\). Note that 60 implies that \[\label{eqn:H-error-1} \pi_{P\#}(T\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_\sigma(\xi)) = q\llbracket B^n_\sigma(\pi_P(\xi))\rrbracket\tag{61}\] for any \(C_\sigma(\xi)\subset C_{2\rho}(0)\).
Now fix any \(\epsilon\in (0,1/2)\). For \(\xi\in A^n_{\rho,\rho/2}\), define \[\bar{\sigma}(\xi):= \sup\left(\{0\}\cup\left\{\sigma^{-n-2}\int_{C_{\sigma}(\xi)}d_P^2\, d\|T\| \geq \chi^2(\sigma/\rho)^{2\epsilon} E\right\}\right),\] for \(\chi^2:= 64^{n+8}\). If \(\sigma \in [\rho/32,\rho/2]\), then we can estimate \[\begin{align} \int_{C_\sigma(\xi)}d_P^2\, d\|T\| \leq \int_{C_{2\rho}(0)}d_P^2\, d\|T\| \leq E(2\rho)^{n+2} & \leq 64^{n+4}(\sigma/\rho)^{2\epsilon}E \sigma^{n+2}\\ & < \chi^2(\sigma/\rho)^{2\epsilon}E\sigma^{n+2} \end{align}\] where the last inequality is strict by our choice of \(\chi\). This shows us that we must have \(\bar{\sigma}(\xi)\leq\rho/32\) for all \(\xi\in A^n_{\rho,\rho/2}\).
For \(\xi \in A^n_{\rho,\rho/2}\) and definition of \(\bar{\sigma}(\xi)\) we have, by Lemma 2, \[\sup_{x\in \mathrm{spt}\|T\|\cap C_{16\sigma}(\xi)}d_P(x) \leq c(n)\chi(\sigma/\rho)^\epsilon E^{1/2}\cdot\sigma \quad \text{for all }\sigma\in [\bar{\sigma}(\xi),\rho/32],\] which in particular implies, if setting \(\Xi:= \{\xi\in A^n_{\rho,\rho/2}:\bar{\sigma}(\xi)>0\}\), that \(\mathrm{spt}\|T\|\cap ((A^n_{\rho,\rho/2}\setminus\Xi)\times\mathbb{R}^m)\subset P\).
Now choose a Vitali collection \(\{\xi_i\}_i\subset\Xi\) so that, writing \(\sigma_i:= \bar{\sigma}(\xi_i)\), the balls \(\{B_{2\sigma_i/5}(\xi_i)\}_i\) are disjoint and the balls \(\{B_{2\sigma_i}(\xi_i)\}_i\) cover \(\Xi\). We then have from Lemma 4 that \[\mathcal{E}_{T,P}(\xi_i,8\sigma_i) \leq c(n,m)E_{T,P}(\xi_i,16\sigma_i) \leq 16 \chi^2(\sigma_i/\rho)^{2\epsilon}E,\] and therefore in view of 61 , provided \(\delta = \delta(n,m,\theta)\) is sufficiently small, we can apply Almgren’s Lipschitz approximation (Theorem 5) to find Lipschitz functions \(u_i:B^n_{2\sigma_i}(\xi_i) \to \mathcal{A}_q(\mathbb{R}^m)\) and subsets \(K_i\subset B^n_{2\sigma_i}(\xi_i)\) satisfying \[\label{eqn:H-error-lip1} \textrm{Lip}(u_i) \leq c(\sigma_i/\rho)^{\gamma\epsilon} E^\gamma;\tag{62}\] \[\label{eqn:H-error-lip2} T\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(K_i\times \mathbb{R}^m) = \mathbf{v}(u_i) \mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}(K_i\times\mathbb{R}^m);\tag{63}\] \[\label{eqn:H-error-lip3} |B^n_{2\sigma_i}(\xi_i)\setminus K_i| + \|T\|((B^n_{2\sigma_i}(\xi_i)\setminus K_i)\times\mathbb{R}^m) \leq c(\sigma_i/\rho)^{2\epsilon(1+\gamma)}E^{(1+\gamma)}\sigma_i^n.\tag{64}\] Here, \(c = c(n,m,\theta)\) and \(\gamma = \gamma(n,m,\theta)>0\). We can then estimate \[\begin{align} \int_{C_{2\sigma_i}(\xi_i)}d_P^2&\|\pi_T-\pi_P\|^2\, d\|T\|\\ &\leq c\int_{K_i\cap B^n_{2\sigma_i}(\xi_i)}|u_i|^2|Du_i|^2\, dx + c\sigma_i^{n+2}(\sigma_i/\rho)^{2\epsilon(2+\gamma)}E^{2+\gamma}\\ & \leq c(\sigma_i/\rho)^{2\gamma\epsilon}E^{\gamma}\int_{K_i\cap B^n_{2\sigma_i}(\xi_i)}|u_i|^2\, dx + c\sigma_i^{n+2}(\sigma_i/\rho)^{2\epsilon(2+\gamma)}E^{2+\gamma}\\ & \leq c(\sigma_i/\rho)^{2\gamma\epsilon}E^{\gamma}\int_{C_{2\sigma_i}(\xi_i)}d_P^2\, d\|T\| + c\sigma_i^{n+2}(\sigma_i/\rho)^{2\epsilon(2+\gamma)}E^{2+\gamma}\\ & \leq c(n,m,\theta)\sigma_i^{n+2}(\sigma_i/\rho)^{2\epsilon(1+\gamma)}E^{1+\gamma}. \end{align}\] Noting that by definition of \(\bar{\sigma}\) we have \[E_{T,P}(\xi_i,\sigma_i) \geq \chi^2(\sigma_i/\rho)^{2\epsilon}E \geq \frac{1}{16}E_{T,P}(\xi_i,16\sigma_i),\] and noting that since \(\xi_i\in C_\rho\setminus C_{\rho/2}\) we can find \(B^n_{\sigma_i/10}(y_i)\subset B^n_{2\sigma_i/5}(\xi_i)\cap A^n_{\rho,\rho/2}\) and then apply Lemma 13 at scale \(C_{2\sigma_i}(\xi_i)\) to deduce \[\begin{align} -\rho\int_{C_{2\sigma_i/5}(\xi_i)}d_P^2\frac{1}{r}\phi^\prime(r/\rho)\, d\|T\| & \geq \int_{C_{\sigma_i/10}(y_i)}d_P^2\, d\|T\|\\ & \geq \frac{1}{c}\int_{C_{16\sigma_i}(\xi_i)}d_P^2\, d\|T\|\\ & \geq\frac{1}{c}\int_{C_{\sigma_i}(\xi_i)}d_P^2\, d\|T\|\\ & \geq \frac{1}{c}\sigma_i^{n+2}(\sigma_i/\rho)^{2\epsilon}E \end{align}\] for \(c = c(n,m,\theta)\). We then compute \[\begin{align} -\rho\int d_P^2\|\pi_T-\pi_P\|^2&\frac{1}{r}\phi^\prime(r/\rho)\, d\|T\|\\ & \leq 4\sum_i\int_{C_{2\sigma_i}(\xi_i)}d_P^2\|\pi_T-\pi_P\|^2\, d\|T\|\\ & \leq c\sum_i\sigma_i^{n+2}(\sigma_i/\rho)^{2\epsilon(1+\gamma)}E^{1+\gamma}\\ & \leq c\sum_i\left(-\rho\int_{C_{2\sigma_i/5}(\xi_i)}d_P^2\frac{1}{r}\phi^\prime(r/\rho)\, d\|T\|\right)\cdot (\sigma_i/\rho)^{2\epsilon\gamma}E^{\gamma}\\ & \leq c(n,m,\theta)E^{\gamma}\left(-\rho\int d_P^2\frac{1}{r}\phi^\prime(r/\rho)\, d\|T\|\right). \end{align}\] Combining the above estimate with \(|\nabla r|^2\geq 1-\|\pi_T-\pi_P\|^2\), and choosing \(\delta = \delta(n,m,\theta)\) sufficiently small, we get \[\label{eqn:H-error-2} -\rho^{1-n}\int d_P^2\frac{1}{r}\phi^\prime(r/\rho)\, d\|T\| \leq (1+cE^{\gamma})H_{T,P}(\rho) \leq 2H_{T,P}(\rho),\tag{65}\] which again when combined with the previous estimate implies 56 . The proof of 57 then follows as in [25]. Finally, the first inequality in 58 is immediate from the definition of \(H_{T,P}(\rho)\), whilst the second follows from 65 . ◻
Lemma 15. Let \(q\in \mathbb{N}\), \(a>0\), and \(T\) be an area-minimizing integral \(n\)-current in \(C_8(0)\) satisfying \[\label{eqn:D-helper-hyp1} (\partial T)\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_8(0) = 0, \quad \pi_{P\#}T = q\llbracket B^n_\infty(0)\rrbracket,\quad \mathcal{E}_{T,P}(0,8)\leq \epsilon^2,\tag{66}\] \[\int_{C_{1/2}(0)}|\vec{T}-\vec{P}|^2\, d\|T\| \geq a^2\int_{C_8(0)}|\vec{T}-\vec{P}|^2\, d\|T\|.\] Then, provided \(\epsilon= \epsilon(n,m,q,a)\in (0,1)\) is sufficiently small, there is \(b = b(n,m,q,a)>0\) so that \[\int_{C_{1/20}(\xi)}\|\pi_T-\pi_P\|^2\, d\|T\| \geq b^2\int_{C_8(0)}|\vec{T}-\vec{P}|^2\, d\|T\| \quad \text{for all }\xi\in B^n_{1/5}(0).\]
Proof. The proof is the same as in [25], using Lemma 12 and Almgren’s Lipschitz approximation (Theorem 5). Note that one does not require \(T\) to be close to \(q\llbracket B^n_1(0)\rrbracket\) in \(L^2\) height to apply Theorem 5, it suffices to assume 66 . ◻
Lemma 16. Let \(T\) be an area-minimizing integral \(n\)-current in \(C_{2\rho}(0)\) with: \[(\partial T)\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_{2\rho}=0, \quad \Theta_T(0,\rho/2)\geq 1/2,\quad \Theta_T(0,2\rho)\leq\theta,\] \[E_{T,P}(0,2\rho)\leq E \leq \delta^2.\] Then, provided \(\delta = \delta(n,m,\theta)\) is sufficiently small, we can find positive constants \(c^\prime<c^{\prime\prime}<c\) and \(\gamma_1<\gamma\) depending only on \(n,m,\theta\) such that \[\begin{align} &\rho^{1-n}\int_{C_\rho(0)}|\vec{T}-\vec{P}|^4\, d\|T\| \label{eqn:D-error-concl1}\\ & \leq c^\prime E^{\gamma-\gamma_1} D_{T,P}(\rho)\frac{d}{d\rho}\left[(\rho^{-2}D_{T,P}(\rho))^{\gamma_1}\right] + cE^{\gamma}\rho^{-1}D_{T,P}(\rho),\nonumber \end{align}\tag{67}\] and, subsequently, \[\begin{align} &\left|D^\prime_{T,P}(\rho) + 2\rho^{-n}\int\frac{|\pi_{P^\perp}(\nabla^T r)|^2}{|\nabla^T r|^2} r\phi^\prime(r/\rho)\, d\|T\|\right| \label{eqn:D-error-concl2} \\ &\leq c^{\prime\prime}E^{\gamma-\gamma_1}D_{T,P}(\rho)\frac{d}{d\rho}\left[(\rho^{-2}D_{T,P}(\rho))^{\gamma_1}\right] + cE^{\gamma}\rho^{-1}D_{T,P}(\rho).\nonumber \end{align}\tag{68}\]
Proof. As in the proof of Lemma 14, we can assume \[\label{eqn:D-error-1} \pi_{P\#}T = q\llbracket B^n_{2\rho}(0)\rrbracket\tag{69}\] for some integer \(1\leq q\leq\theta\). From Lemma 4 we have \[\mathcal{E}_{T,P}(0,\rho) \equiv \frac{1}{2}\rho^{-n}\int_{C_\rho(0)}|\vec{T}-\vec{P}|^2\, d\|T\| \leq c(n,m)E.\] Fix any \(\epsilon\in (0,1/2)\) and set \(\chi^2:= 64^{n+8}\). For each \(\xi\in B^n_\rho\), define \[\bar{\sigma}(\xi):= \sup\left(\{0\}\cup \left\{\sigma\in (0,\rho/2]:\sigma^{-n}\int_{C_{\sigma}(\xi)}|\vec{T}-\vec{P}|^2\, d\|T\| \geq \chi^2(\sigma/\rho)^{2\epsilon}E\right\}\right).\] Just as in Lemma 14, our choice of \(\chi\) gives \(\bar{\sigma}(\xi)\leq\rho/32\) for all \(\xi\in B_\rho^n\).
Set \(\Xi:= \{\xi\in B^n_\rho:\bar{\sigma}(\xi)>0\}\). Noting that for any \(\xi\in B^n_\rho\) we have, by definition of \(\bar{\sigma}(\xi)\), \[\label{eqn:D-error-2} \mathcal{E}_{T,P}(\xi,\sigma)\leq c(n)(\sigma/\rho)^{2\epsilon}E \quad \text{for all }\sigma\in [\bar{\sigma}(\xi),\rho/2],\tag{70}\] we see that in particular \(\vec{T}(x) = \vec{P}\) for \(\|T\|\)-a.e. \(x\in (B^n_\rho\setminus \Xi)\times\mathbb{R}^m\).
Choose a Vitali collection \(\{\xi_i\}_i\subset\Xi\) so that, writing \(\sigma_i := \bar{\sigma}_i(\xi_i)\), the balls \(\{B_{2\sigma_i}(\xi_i)\}_i\) form a cover of \(\Xi\) and the balls \(\{B_{2\sigma_i/5}(\xi_i)\}_i\) are disjoint. From 70 and 69 , we can apply Almgren’s Lipschitz approximation (Theorem 5) in each cylinder \(C_{2\sigma_i}(\xi_i)\) to obtain Lipschitz functions \(u_i:B^n_{2\sigma_i}(\xi_i)\to \mathcal{A}_q(\mathbb{R}^m)\) and subsets \(K_i\subset B^n_{2\sigma_i}(\xi_i)\) satisfying the same estimates as in 62 , 63 , 64 .
We now estimate \[\begin{align} \int_{C_{2\sigma_i}(\xi_i)}&|\vec{T}-\vec{P}|^4\, d\|T\|\\ &\leq c\int_{K_i\cap B^n_{2\sigma_i}(\xi_i)}|Du_i|^4\, dx + c\sigma_i^n(\sigma_i/\rho)^{2\epsilon(1+\gamma)}E^{1+\gamma}\\ & \leq c(\sigma_i/\rho)^{2\epsilon\gamma}E^{\gamma}\int_{K_i\cap B^n_{2\sigma_i}(\xi_i)}|Du_i|^2\, dx + c\sigma_i^n(\sigma_i/\rho)^{2\epsilon(1+\gamma)}E^{1+\gamma}\\ & \leq c\sigma_i^n(\sigma_i/\rho)^{2\epsilon(1+\gamma)}E^{1+\gamma}. \end{align}\] Since \(\xi_i\in B^n_\rho\), we can choose \(B^n_{\sigma_i/5}(y_i)\subset B^n_{2\sigma_i/5}(\xi_i)\cap B_\rho^n\), and then on \(B^n_{\sigma_i/10}(y_i)\) we have \(\phi(r/\rho)\geq \sigma_i/(20\rho)\). Using Lemma 15 in a similar manner to the proof of Lemma 14, we deduce \[\begin{align} (\sigma_i/(20\rho))^{-1}\int_{C_\rho(0)}\|\pi_T-\pi_P\|^2\phi(r/\rho)\, d\|T\| & \geq \int_{C_{\sigma_i/10}(y_i)}\|\pi_T-\pi_P\|^2\, d\|T\|\\ & \geq \frac{1}{c}\int_{C_{16\sigma_i}(\xi_i)}|\vec{T}-\vec{P}|^2\, d\|T\|\\ & \geq\frac{1}{c}(\sigma_i/\rho)^{2\epsilon}E\sigma_i^n \end{align}\] for suitable \(c = c(n,m,\theta)\). In particular, for each \(i\) we have \[\rho^{-2}D_{T,P}(\rho)\geq \frac{1}{c}(\sigma_i/\rho)^{n+1+2\epsilon}E.\] We then estimate \[\begin{align} \int_{C_\rho(0)}&|\vec{T}-\vec{P}|^4\, d\|T\|\nonumber\\ & \leq \sum_i \int_{C_{2\sigma_i}(\xi_i)}|\vec{T}-\vec{P}|^4\, d\|T\|\nonumber\\ & \leq c\sum_i\sigma_i^n(\sigma_i/\rho)^{2\epsilon(1+\gamma)}E^{1+\gamma}\nonumber\\ & \leq c\sum_i (\sigma_i/\rho)^{2\epsilon\gamma}E^{\gamma}\int_{C_{2\sigma_i/5}(\xi_i)\cap C_\rho(0)}\|\pi_T-\pi_P\|^2\, d\|T\|\nonumber\\ & = c\sum_i\big((\sigma_i/\rho)^{n+1+2\epsilon}E\big)^{\gamma_1}E^{2\gamma-2\gamma_1}\int_{C_{2\sigma_i/5}(\xi_i)\cap C_\rho(0)}\|\pi_T-\pi_P\|^2\, d\|T\|\nonumber\\ & \leq c(\rho^{-2}D_{T,P}(\rho))^{\gamma_1}E^{\gamma-\gamma_1}\int_{C_\rho(0)}\|\pi_T-\pi_P\|^2\, d\|T\|,\label{eqn:D-error-estimate-1} \end{align}\tag{71}\] where \(\gamma_1:= \frac{2\epsilon\gamma}{n+1+2\epsilon} < \gamma\) and \(c = c(n,m,\theta)>0\).
Now we have (by definition of \(\phi\)) \[\begin{align} \frac{1}{2}\int_{C_\rho(0)}\|\pi_T-\pi_P\|^2\, d\|T\| & \leq \frac{1}{2}\int_{C_\rho(0)}\|\pi_T-\pi_P\|^2\big(\phi(r/\rho)-(r/\rho)\phi^\prime(r/\rho)\big)\, d\|T\|\\ & = (n-1)\rho^{n-2}D_{T,P}(\rho) + \rho^{n-1}D_{T,P}^\prime(\rho), \end{align}\] and also note that by Lemma 3 (and Lemma 2 to pass from balls to cylinders) we have \[\rho^{-2}D_{T,P}(\rho) \leq cE,\] and so combining everything (writing \(D := D_{T,P}(\rho)\) as a shorthand), we get \[\begin{align} \rho^{1-n}\int_{C_\rho(0)}|\vec{T}-\vec{P}|^4\, d\|T\| & \leq c_1 E^{\gamma-\gamma_1}(\rho^{-2}D)^{\gamma_1}\big((n-1)\rho^{-1}D+D^\prime\big)\\ & = c_1E^{\gamma-\gamma_1}\rho^{-2\gamma_1}D^{\gamma_1}\big(-2\rho^{-1}D+D^\prime\big)\\ &+ c_1 E^{\gamma-\gamma_1}(\rho^{-2}D)^{\gamma_1}(n+1)\rho^{-1}D\\ & \leq \frac{c_1}{\gamma_1}E^{\gamma-\gamma_1}\rho^{-2\gamma_1}D^{\gamma_1}\big(-2\gamma_1 \rho^{-1}D + \gamma_1 D^\prime\big)\\ &+ cE^{\gamma}\rho^{-1}D\\ & = c^\prime E^{\gamma-\gamma_1} D\frac{d}{d\rho}\big[(\rho^{-2}D)^{\gamma_1}\big] + cE^{\gamma}\rho^{-1}D \end{align}\] This proves 67 . The proof of 68 follows from 67 in the same fashion as in [25]. ◻
Proof of Theorem 7. Combine Lemma 14 and Lemma 16 as in [25] to get: \[\begin{align} \frac{d}{d\rho}\log N_{T,P}(\rho) & = \frac{H_{T,P}(\rho)D^\prime_{T,P}(\rho) - H^\prime_{T,P}(\rho)D_{T,P}(\rho)}{H_{T,P}(\rho)D_{T,P}(\rho)}\\ & \geq -c\rho^{-1}E^{\gamma}-c^\prime E^{\gamma-\gamma_1}\frac{d}{d\rho}\big[(\rho^{-2}D_{T,P}(\rho))^{\gamma_1}\big]. \qedhere \end{align}\] ◻
Finally, we prove Corollary 1.
Proof of Corollary 1. Conclusion ?? follows from 20 . Indeed, we get \[(2N_1-c\epsilon^{2\gamma})\rho^{-1}\leq \frac{d}{d\rho}\log H_{T,P}(\rho) \leq (c\epsilon^{2\gamma} + 2N_2)\rho^{-1}\] and so integrating gives ?? . As for ?? , we follow [25]. Let \(k\) be the largest integer \(\leq \log_2(\rho_2/\rho_1)\). We then compute (using 58 ) \[\begin{align} \int_{C_{\rho_2}}d_P^2\, d\|T\| & \leq \int_{C_{\rho_1}}d_P^2\, d\|T\| + \sum_{i=0}^k\int_{C_{2^{-i}\rho_2}\setminus\overline{C_{2^{-i-1}\rho_2}}}d_P^2\, d\|T\|\\ & \leq \int_{C_{\rho_1}}d_P^2\, d\|T\| + \sum^k_{i=0}(2^{-i}\rho_2)^nH_{T,P}(2^{-i}\rho_2)\\ & \leq \int_{C_{\rho_1}}d_P^2\, d\|T\| + \sum_{i=0}^k (2^{-i}\rho_2/\rho_1)^{2N_2+c\epsilon^{2\gamma}}(2^{-i}\rho_2)^nH_{T,P}(\rho_1)\\ & \leq \left(1+4\sum^\infty_{i=0}(2^{-i})^{2N_2+c\epsilon^{2\gamma}+n}\right)(\rho_2/\rho_1)^{2N_2+c\epsilon^{2\gamma}+n}\int_{C_{\rho_1}}d_P^2\, d\|T\|\\ & \leq \frac{8}{1-2^{-n}}(\rho_2/\rho_1)^{2N_2+c\epsilon^{2\gamma}+n}\int_{C_{\rho_1}}d_P^2\, d\|T\|, \end{align}\] which completes the proof. ◻
Krummel–Wickramasekera have communicated to us that in their forthcoming work [40], [41] they establish refined \(L^2-L^\infty\) estimates that generalize Theorem 3 to area-minimizing currents close to disjoint unions of \(C^2\) graphs defined over a plane or, more generally, over a center manifold. While these generalizations are not required in the present work, for the reader’s convenience we record here the planar version of their theorem (we refer the reader to [41] for the center manifold version).
Fix integers \(1\leq p \leq s\), constants \(\epsilon\in (0,1)\), \(\kappa\in [0,\infty)\), and write \(P = \mathbb{R}^n\times\{0^m\}\subset\mathbb{R}^{n+m}\). Consider functions \(\phi:B_1\to \mathcal{A}_s(\mathbb{R}^k)\) given by \(\phi = \sum^p_{i=1}s_i\llbracket \phi_i\rrbracket\), where \(s_1,\dotsc,s_p\in \mathbb{N}\) satisfy \(\sum_{i=1}^ps_i=s\) and \(\phi_i:B_1\to \mathbb{R}^m\) are \(C^2\) such that \(\|D\phi_i\|_\infty+\|\Delta\phi_i\|_\infty<\epsilon\) and \(\sup|D\phi_i-D\phi_j|\leq \kappa\inf|\phi_i-\phi_j|\) for all \(i,j\).
Theorem 10 (Improved \(L^2-L^\infty\) estimate; [40]). Fix \(Q\in \mathbb{N}\) and integers \(1\leq p\leq s\). Then, for all \(\gamma\in (0,1)\) and \(\kappa\in [0,\infty)\), there exists \(\epsilon= \epsilon(n,m,Q,s,\gamma,\kappa)\in (0,1)\) such that if \(T\) is an \(n\)-dimensional area-minimizing integral current in \(C_1(0)\equiv B^n_1(0)\times\mathbb{R}^k\) such that \[(\partial T)\mathbin{\vrule height 1.3ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.0ex}C_1(0) = 0, \quad \sup_{x\in \mathrm{spt}\|T\|}\mathrm{dist}(x,P)<\infty,\quad \pi_{P\#}T = Q\llbracket B_1\rrbracket,\] \[\mathcal{E}\equiv \mathcal{E}_{T,P}(0,1) < \epsilon^2,\] and if \(\phi\) is as above for these choices of \(s,p,\kappa,\epsilon\), we then have \[\sup_{x\in C_\gamma(0)\cap \mathrm{spt}\|T\|}\mathrm{dist}(x, \mathrm{graph}(\phi)) \leq cE^{1/2}_*\] where \[E_* = \int_{C_1}\mathrm{dist}^2(x,\mathrm{graph}(\phi))\, d\|T\|(x) + \sum^s_{\ell=1}\|D\phi_\ell\|_{\infty}\mathcal{E}+ \sum^s_{\ell=1}\|\Delta\phi_\ell\|^2_\infty + \mathcal{E}^{1+\gamma}.\] Here, \(c = c(n,m,Q,s,\gamma)\in (0,\infty)\) and \(\gamma = \gamma(n,m,Q)\in (0,1)\).