1 Introduction↩︎

The Plateau problem asks whether a given boundary bounds a minimal surface with least area. In the early 1930’s, Douglas [1] and Radó [2] gave the positive answer for Jordan curves in \(\mathbb{R}^3\), and their work was generalized to Riemannian manifolds in 1948 by Morrey [3]. Later, Federer and Fleming [4] introduced the notion of area-minimizing integral currents¹, a well-known generalization of minimal surfaces, to formulate and address the oriented Plateau problem. Due to its broad applicability across all dimensions and codimensions, Federer and Fleming’s theory marks a major success in establishing existence results.

A natural question is whether solutions to the oriented Plateau problem are unique. In general, the uniqueness of solutions to this problem cannot be guaranteed, even with smooth boundary data (see e.g. [5], [6]). Fortunately, uniqueness is known to be a generic property. Morgan [7] first showed that almost every curve in \(\mathbb{R}^3\) bounds a unique area-minimizing surface. In [8], [9], he extended this result to elliptic integrands in arbitrary dimensions and, in the special case of area-minimizing flat chains modulo 2, to all codimensions. These results are confined to the Euclidean setting or require uniform convexity assumptions on the boundary, as the proofs rely on Allard’s boundary regularity theorem [10].

More recently, Caldini, Marchese, Merlo, and Steinbrüchel [11]—based on the boundary regularity theory developed by De Lellis, De Philippis, Hirsch, and Massaccesi [12]—established the generic uniqueness of area-minimizing integral currents in arbitrary ambient manifolds, all dimensions and codimensions, and without the assumption of convexity on the boundary. However, given a specific boundary, it remains unclear whether it uniquely bounds an area-minimizing integral current. Uniqueness can be proved only in restricted cases, such as for surfaces in \(\mathbb{R}^3\) bounded by special Jordan curves [2], [13], for hypersurfaces in \(\mathbb{R}^n\) satisfying suitable conditions [14], [15], or for graphical hypersurfaces over convex domains [16].

In this paper, we prove uniqueness in the oriented Plateau problem under a natural geometric condition, without imposing any restrictions on dimension and codimension. Specifically, we show that every compactly supported calibrated integral current with connected \(C^{3,\alpha}\) boundary uniquely solves the oriented Plateau problem for its boundary data. Our result also recovers several known cases—for instance, \(C^{3,\alpha}\) graphical hypersurfaces over convex domains, since they are calibrated (see 2). As in [11], the conclusion follows from the boundary regularity theory in [12]. However, our method differs in that it relies on unique continuation principles for elliptic operators. Unless otherwise stated, we make the following assumptions:

We can now state our main theorem, which is a special case of 27.

The idea is to derive 2 as a consequence of uniqueness in the Cauchy problem for elliptic operators. Let \(\Omega\subset \mathbb{R}^k\) be a bounded \(C^{1,1}\) domain and let \(L : H^1(\Omega; \mathbb{R}^{n-k}) \rightarrow \mathcal{L}^2(\Omega; \mathbb{R}^{n-k})\) be the second-order principally diagonal² elliptic partial differential operator: \[(Lu(\mathbf{x}))^\sigma= \partial_i(a^{ij}(\mathbf{x})u_{x^j}^\sigma(\mathbf{x})) + b_\gamma^{\sigma s}(\mathbf{x}) u_{x^s}^\gamma(\mathbf{x}) + c_\gamma^\sigma(\mathbf{x}) u^\gamma(\mathbf{x}) \text{ for } \sigma= 1,\ldots, n-k, \label{L1}\tag{1}\] where \(A := (a^{ij})_{1 \leq i,j\leq k} \in C^{0,1}(\overline{\Omega}; \mathbb{R}^{k\times k})\) is a positive definite symmetric matrix with eigenvalues in \([\lambda, \Lambda]\) for \(0 < \lambda < \Lambda < \infty\) and \(b_\gamma^{\sigma s}, c_\gamma^\sigma\in \mathcal{L}^\infty(\Omega)\). The Cauchy problem asks whether there exists a unique \(u \in H^1(\Omega; \mathbb{R}^{n-k})\) solving \[\label{cauchy1} \begin{cases} Lu(\mathbf{x}) = 0 \text{ in } \Omega\\ u(\mathbf{x}) = 0 \text{ and } \partial_\nu u(\mathbf{x}) = 0 \text{ on } \Gamma, \end{cases}\tag{2}\] where \(\Gamma\) is a relatively open portion of \(\partial\Omega\), \(\nu\) is the unit outer normal to \(\Gamma\), and \(L\) is given by 1 . Since any solution \(u \in H^1(\Omega; \mathbb{R}^{n-k})\) to 2 has zero trace on \(\Gamma\), it is \(C^{1,\alpha}\) up to \(\Gamma\) by elliptic regularity so that \(u\) and \(\partial_\nu u\) continuously vanish along \(\Gamma\). By boundary Carleman estimates, if \(u\) solves 2 then \(u \equiv 0\) in \(\Omega\) (see [17], [18] and [19]).

Since uniqueness in the Cauchy problem follows from Carleman estimates, it is intimately tied to the unique continuation properties of elliptic operators. For this reason, it is often called unique continuation from Cauchy data. We say that a function \(u \in H^1(\Omega; \mathbb{R}^{n-k})\) has the strong unique continuation property (SUCP) in \(\Omega\) if \(u \equiv 0\) whenever there is a point \(\mathbf{x}_0 \in \Omega\) at which \(u\) vanishes to infinite order [20]. The SUCP for solutions \(u\) to \(Lw = 0\) in \(\Omega\) with \(L\) in 1 was first proved by Aronszajn, Krzywicki, and Szarski in [18] using Carleman estimates (see [19] and [20]–[22]). The condition that \(L\) is principally diagonal is necessary, since there are counterexamples to SUCP and uniqueness in 2 when we allow for leading-order coupling (see e.g. [23]). Moreover, the results in [18]–[20], [22] are essentially sharp, because for each \(\alpha \in (0,1)\) there are scalar divergence form and non-divergence form elliptic operators with \(C^{0,\alpha}\) leading coefficient functions for which SUCP fails (see [23], [24]).

The first step in the proof of 2 is to prove uniqueness in the Cauchy problem for Lipschitz stationary graphs. Let \(\Omega\subset \mathbb{R}^k\) be a domain (not necessarily bounded) with Lipschitz boundary and suppose \(u \in C_{\text{loc}}^{0}(\overline{\Omega}; \mathbb{R}^{n-k}) \cap C_{\text{loc}}^{0,1}(\Omega; \mathbb{R}^{n-k})\). Then its graph \(\Sigma_u\) is a \(k\)-dimensional Lipschitz submanifold of \(\mathbb{R}^n\) with boundary \[\partial\Sigma_u := \{(\mathbf{x},u(\mathbf{x})) : \mathbf{x}\in \partial\Omega\subset \mathbb{R}^k\} \subset \mathbb{R}^n.\] Define \[F: \mathbb{R}^{(n-k)\times k} \rightarrow \mathbb{R}\text{ by } F(\mathbf{p}) := \sqrt{\det( I + \mathbf{p}^T\mathbf{p})}.\] Then the \(k\)-dimensional surface area of \(\Sigma_u\) is given by the formula \[\label{areafunctional} \mathcal{A}(Du) := \int_{\Sigma_u} \, d\mathcal{H}^k = \int_\Omega\sqrt{\det g(Du(\mathbf{x}))}\, d\mathbf{x}= \int_\Omega F(Du(\mathbf{x})) \, d\mathbf{x},\tag{3}\] where \(\mathcal{H}^k\) is the \(k\)-dimensional Hausdorff measure and the induced metric \(g\) is \(\mathcal{H}^k\)-a.e. defined by \[g(Du(\mathbf{x})) := I + Du(\mathbf{x})^TDu(\mathbf{x}) = (\delta_{ij} + u_{x^i}(\mathbf{x}) \cdot u_{x^j}(\mathbf{x}))_{1 \leq i,\,j \leq k}. \label{metric}\tag{4}\] The submanifold \(\Sigma_u\) can be identified with a stationary varifold \(V_u:= \underline{v}(\Sigma_u, 1)\) if and only if \(u\) is a stationary solution to the minimal surface system (MSS) in \(\Omega\): \[\begin{cases} \partial_i(F_{p_i^\gamma}(Du(\mathbf{x}))u_{x^j}^\gamma(\mathbf{x}) - F(Du(\mathbf{x}))\delta_{ij}) = 0 \text{ for } j = 1,\ldots, k \\ \partial_i(F_{p_i^\sigma}(Du(\mathbf{x}))) = 0 \text{ for } \sigma= 1,\ldots, n-k, \end{cases} \label{MSS}\tag{5}\] where \(DF(\mathbf{p}) := (F_{p_i^\sigma}(\mathbf{p}))_{1 \leq i \leq k}^{1 \leq \sigma \leq n-k}\) is viewed as a map \(\mathbb{R}^{(n-k) \times k} \rightarrow \mathbb{R}^{(n-k) \times k}\). The first equation in 5 is called the inner variation system for \(\mathcal{A}\), while the second is called the outer variation system, or Euler–Lagrange system, for \(\mathcal{A}\).

When \(u \in C_{\text{loc}}^2(\Omega; \mathbb{R}^{n-k}) \cap C_{\text{loc}}^{0}(\overline{\Omega}; \mathbb{R}^{n-k})\), the MSS is equivalent to the following quasi-linear elliptic system in non-divergence form: \[g^{ij}(Du(\mathbf{x}))u_{x^i x^j}^\sigma(\mathbf{x}) = 0 \text{ for each } \sigma = 1,\ldots, n-k \text{ in } \Omega, \label{MSS2}\tag{6}\] where we have set \(g^{-1} := (g^{ij})_{1 \leq i,j \leq k}\). Note that \(g^{-1}\) is positive definite with eigenvalues in \([\lambda, \Lambda] \subset (0,\infty)\) for some numbers \(\lambda, \Lambda\) depending on the Lipschitz constant for \(u\). Any solution \(u\) solving 6 is called a classical solution to the MSS. When \(k = n-1\) (i.e. \(\Sigma_u\) has codimension one), the outer variation equation is the minimal surface equation (MSE), which is the quasi-linear divergence form scalar equation with coefficients \[a^{ij}(Du(\mathbf{x})) := \frac{1}{\sqrt{1 + |Du(\mathbf{x})|^2}}\Bigg(\delta_{ij} - \frac{u_{x^i}(\mathbf{x}) u_{x^j}(\mathbf{x})}{\sqrt{1 + |Du(\mathbf{x})|^2}}\Bigg). \label{MSE}\tag{7}\]

Now, a straightforward computation using the fundamental theorem of calculus (FTC) shows that the difference of two solutions to 6 (or 7 ) solves a homogeneous equation for a principally diagonal operator of the form 1 on a neighborhood of \(\Gamma\) in \(\Omega\). Applying this along with the SUCP and partial regularity for Lipschitz stationary solutions to the MSS [25], we obtain uniqueness in the Cauchy problem for 5 .

Step 2 in the proof of 2 is to formulate and prove unique continuation from Cauchy data in the general setting of area-minimizing integral currents. However, the difficulty lies in regularity: In general, area-minimizing currents tend to exhibit singularities. When the codimension is one, it is well known that, away from its boundary, the support of an area-minimizing integral \(k\)-current in Euclidean space is an analytic hypersurface outside of a relatively closed subset of Hausdorff dimension at most \(k-7\). We refer to this set as the interior singular set. In higher codimension, Almgren [26] showed that the interior singular set of area-minimizing integral \(k\)-currents in Euclidean space has Hausdorff dimension at most \(k-2\).

Boundary regularity is more subtle. In codimension one, several results are known (see e.g. [10], [27]). On the other hand, in higher codimension our understanding remained limited for a long time. Only recently did De Lellis, De Philippis, Hirsch, and Massaccesi [12] establish the first general boundary regularity theorem without restrictions on the codimension or the ambient manifold. With the help of their regularity theory, we are able to prove unique continuation from Cauchy data for compactly supported area-minimizing integral currents. In the setting of area-minimizing currents, this can be stated as follows:

Due to the boundary regularity, we can choose a point \(p \in \Gamma^\prime\) such that, near \(p\), both \(\mathop{\mathrm{spt}}T\) and \(\mathop{\mathrm{spt}}T^\prime\) can be represented as Lipschitz stationary graphs over a domain whose boundary contains a relatively open \(C^{3, \alpha}\) portion. Moreover, their defining functions satisfy the same Cauchy data, so 3 and 4 together imply that \(\mathop{\mathrm{spt}}T\) and \(\mathop{\mathrm{spt}}T^\prime\) agree in a neighborhood of \(p\). Since the interior regular sets of \(T\) and \(T^\prime\) are analytic, a unique continuation argument (i.e. 19) shows that \(T\) and \(T^\prime\) share the same regular part. Incorporating a discussion of the multiplicities yields the conclusion of 5.

The third and final step is to show that, for a given boundary \(\Gamma\subset \mathbb{R}^n\) satisfying the hypotheses in 1, the existence of a calibrated current \(T \in \mathcal{I}_{k,c}(\mathbb{R}^n)\) with \(\partial T = [[\Gamma]]\) forces any other area-minimizing current \(T^\prime \in \mathcal{I}_{k,c}(\mathbb{R}^n)\) with \(\partial T^\prime = [[\Gamma]]\) to share the same Cauchy data in the sense of 5 along a relatively open portion of \(\Gamma\). The argument follows from a couple of observations. The first is that \(T^\prime\) is also calibrated by \(\varphi\) (26). Then, the first cousin principle (24) together with the boundary regularity theory imply that the true tangent spaces of \(\mathop{\mathrm{spt}}T\) and \(\mathop{\mathrm{spt}}T^\prime\) coincide on a relatively open portion of \(\Gamma\). This allows us to apply 5, from which uniqueness follows. We conclude this discussion with a remark on general ambient manifolds.

The organization of this paper is as follows. Section 2 introduces the basic notation and definitions concerning area-minimizing and calibrated integral currents. In Section 3, we briefly recall the interior regularity theory for Lipschitz stationary solutions to the MSS 5 , and then present the proof of 3. We also record certain boundary regularity results from [12] and use them to prove 5. Following this, we discuss the generalization to area-minimizing currents with unbounded support or higher multiplicity. Section 4 contains a general version of 2 (i.e. 27) and the sharpness of the assumptions in 27. Section 5 discusses 6 in depth. Finally, in Appendix A we prove the following folklore result: If a compactly supported area-minimizing integral current is extendable in a suitable sense, then it is unique. We compare this with 2.

2 Preliminaries↩︎

2.1 Notation and basic definitions↩︎

For fixed \(n \in \mathbb{N}\), we define \(\Lambda^1(\mathbb{R}^n) := (\mathbb{R}^n)^*\). That is, \(\Lambda^1(\mathbb{R}^n)\) is the space of linear functionals \(\ell: \mathbb{R}^n \rightarrow \mathbb{R}\). For each \(k \geq 2\), we will write \(\Lambda^k(\mathbb{R}^n)\) for the space of alternating \(k\)-linear functions (i.e. \(k\)-covectors). The space of smooth \(k\)-forms on \(\mathbb{R}^n\) is denoted by \(\mathcal{E}^k(\mathbb{R}^n) \mathrel{\vcenter{:}}= C^\infty(\mathbb{R}^n; \Lambda^k(\mathbb{R}^n))\). We will write \(\mathcal{D}^k(\mathbb{R}^n)\) for the space of smooth \(k\)-forms with compact support, which is a subspace of \(\mathcal{E}^k(\mathbb{R}^n)\) but equipped with the usual locally convex topology. The space of \(k\)-vectors is denoted \(\Lambda_k(\mathbb{R}^n)\). A \(k\)-vector is called simple if \(\xi = v_1 \wedge v_2 \wedge\cdots \wedge v_k\), where \(v_i \in \Lambda_1(\mathbb{R}^n)\) for all \(i = 1, \ldots, k\) and \(\{v_1,\ldots, v_k\}\) forms a linearly independent set.

Note that we can identify the oriented linearly independent set \(\langle v_1, \ldots, v_k \rangle\) with an oriented \(k\)-plane in \(\mathbb{R}^n\) passing through the origin. In the same vein, such a \(k\)-plane in \(\mathbb{R}^n\) can be represented by the simple \(k\)-vector formed as a wedge product of all elements in its oriented basis. That is, a simple \(k\)-form is equivalent to an oriented \(k\)-plane passing through the origin, up to a scalar factor. We will therefore write \[G(k, \mathbb{R}^n) := \{\xi \in \Lambda_k(\mathbb{R}^n): |\xi| = 1 \text{ and } \xi \text{ is simple}\}\] for the Grassmanian of oriented \(k\)-planes in \(\mathbb{R}^n\) through the origin.

The comass norm of \(\varphi\in \mathcal{\mathcal{E}}^k(\mathbb{R}^n)\) at \(p \in \mathbb{R}^n\) is given by \[||\varphi||_p := \sup_{\xi \in G(k,\mathbb{R}^n)} \langle \varphi_p, \xi \rangle, \label{cmassp}\tag{8}\] where \(\langle \cdot, \cdot \rangle\) is the dual pairing³ for \(\Lambda^k(\mathbb{R}^n)\) and \(\Lambda_k(\mathbb{R}^n)\). The comass norm of \(\varphi\in \mathcal{E}^k(\mathbb{R}^n)\), denoted \(||\varphi||\), is the supremum in 8 over \(p \in \mathbb{R}^n\). We say \(\varphi\) has comass one if \(||\varphi||=1\).

2.2 Integral Currents↩︎

The space \(\mathcal{D}_k(\mathbb{R}^n)\) will represent the continuous dual space of \(\mathcal{D}^k(\mathbb{R}^n)\) (i.e. the \(k\)-currents). For any \(T \in \mathcal{D}_k(\mathbb{R}^n)\), we define \(\partial T \in \mathcal{D}_{k-1}(\mathbb{R}^n)\) (i.e. the boundary of \(T\)) by \(\partial T(\varphi) := T(d\varphi) \text{ for all } \varphi\in \mathcal{D}^{k-1}(\mathbb{R}^n)\). For \(T \in \mathcal{D}_k(\mathbb{R}^n)\), the mass of \(T\) is given by \[\mathbb{M}(T) := \sup_{\varphi\in \mathcal{D}^k(\mathbb{R}^n)}\{T(\varphi): ||\varphi|| \leq 1 \}.\] If \(U \subset \mathbb{R}^n\) is an open set, we can define the mass of \(T\) in \(U\), denoted by \(\mathbb{M}_U(T)\), by instead taking the supremum over those \(\varphi\) with \(\mathop{\mathrm{spt}}\varphi\subset U\). We say \(T\) has finite mass if \(\mathbb{M}(T) < \infty\), and \(T\) has locally finite mass if \(\mathbb{M}_W(T) < \infty\) for every open \(W \subset\subset \mathbb{R}^n\). If both \(T\) and \(\partial T\) have (locally) finite mass, then \(T\) is called (locally) normal.

The function \(\theta_T\) is the multiplicity function for \(T\) and \(\xi_T\) is called the orientation for \(T\). We will write \(\|T\| \mathrel{\vcenter{:}}= \mathcal{H}^k \llcorner \theta_T \llcorner M_T\) for the mass measure of \(T\). Note that \(\mathbb{M}_U(T) = \|T\|(U)\) for every Borel set \(U \subset \mathbb{R}^n\). Moreover, \(\mathop{\mathrm{spt}}T = \mathop{\mathrm{spt}}\|T\|\), where \(\mathop{\mathrm{spt}}\|T\|\) is the support of \(\|T\|\) in the usual sense of Radon measures. We shall write \(\mathcal{I}_k(\mathbb{R}^n)\) for the set of all integral \(k\)-currents. The subset of currents in \(\mathcal{I}_k(\mathbb{R}^n)\) with compact support is denoted \(\mathcal{I}_{k,c}(\mathbb{R}^n)\).

Federer and Fleming show that the homology of the chain complex of currents with compact support is related to the singular homology [4]. As an important consequence, we have:

2.3 Area-minimizing integral currents↩︎

Of particular interest in the present paper are the area-minimizing integral \(k\)-currents.

Throughout the paper, we will often consider the case when \(T\) is a globally area-minimizing integral current and has non-zero boundary \(\partial T := [[\Gamma]]\), where \(\Gamma\subset \mathbb{R}^n\) satisfies 1. If \(T\) has compact support, then \(T\) has the least mass among all compactly supported integral currents with boundary \([[\Gamma]]\). If \(T\) is not compactly supported, then \(T\) has the least mass among all “compact perturbations” of its support.

2.3.1 Rectifiable Varifolds↩︎

Let \(M \subset \mathbb{R}^n\) be a \(k\)-rectifiable Borel set and \(\theta\in \mathcal{L}^1_{loc}(M; d\mathcal{H}^k)\). A rectifiable \(k\)-varifold \(\underline{v}(M, \theta)\) is the equivalence class of all pairs \((M^\prime, \theta^\prime)\), where \(M^\prime\) is \(k\)-rectifiable with \(\mathcal{H}^k(M \Delta M^\prime) = 0\) and \(\theta= \theta^\prime\) \(\mathcal{H}^k\)-a.e. on \(M \cap M^\prime\). In particular, given an integral current \(T\), the varifold associated with \(T\) is just the equivalence class \(\underline{v}(M_T, \theta_T)\), i.e. forgetting the orientation \(\xi_T\). A rectifiable varifold is called stationary if the first variation vanishes. Loosely speaking, stationary rectifiable varifolds can be thought of as Lipschitz minimal submanifolds (not necessarily oriented). We refer the interested reader to [28] for more on rectifiable varifolds.

2.3.2 Densities↩︎

Let \(U \subset \mathbb{R}^n\) be an open set, and \(p \in U\). For \(T \in \mathcal{I}_{k}(\mathbb{R}^n)\) that is area-minimizing in \(U\) and satisfies \(\mathop{\mathrm{spt}}T \subset U\), the density of \(T\) at \(p\) is defined by \[\Theta_T(p) = \lim_{r \rightarrow 0} \frac{\|T\|(B^n_r(p))}{\omega_k r^k} \label{dense}\tag{9}\] whenever the latter limit exists. Here, \(\omega_k\) denotes the volume of the unit \(k\)-ball. Whenever \(p \notin \mathop{\mathrm{spt}}(\partial T)\), the existence of this limit is always guaranteed by the monotonicity formula (cf. [28]), since the varifold associated with an area-minimizing current is stationary (cf. [28]). Thus, area-minimizing integral currents can be heuristically thought of as oriented Lipschitz minimal submanifolds having the least surface area (i.e. mass) with respect to their boundaries. The monotonicity formula also shows that \(\Theta_T(p)\) is an upper semi-continuous function of \(p\). We finally note that \(\Theta_T(p) = \theta_T(p)\) for \(\mathcal{H}^k\)-a.e. \(p \in \mathop{\mathrm{spt}}T\) (cf. [28]). Therefore, \(T\) has the canonical representation \[\label{cano} T = \underline{\tau}(\mathop{\mathrm{spt}}T, \Theta_T, \xi_T).\tag{10}\]

2.4 Calibrated integral currents↩︎

Let \(\varphi\in \mathcal{E}^k(\mathbb{R}^n)\) have comass one. For \(p \in \mathbb{R}^n\), we write \[\mathcal{G}_p(\varphi) \mathrel{\vcenter{:}}= \{\xi \in G(k, \mathbb{R}^n): \langle \varphi_p, \xi \rangle = 1\}\] for the collection of planes where \(\varphi_p\) achieves its maximum. Such \(\mathcal{G}_p(\varphi)\) is called the contact set of \(\varphi\) at \(p\). We also define \[\mathcal{G}(\varphi) \mathrel{\vcenter{:}}= \bigcup_{p \in \mathbb{R}^n} \mathcal{G}_p(\varphi).\]

Calibrated currents achieve their mass when acting on the calibration, and are area-minimizing.

There are many examples of calibrations and calibrated currents, the easiest example being the current associated to the coordinate plane \(\mathbb{R}^k \times \{0\} \subset \mathbb{R}^{k} \times \mathbb{R}^{n-k}\) calibrated by \(\varphi(\mathbf{x}) := dx^1 \wedge \cdots \wedge dx^k\), \(\mathbf{x}:= (x^1, \ldots, x^n) \in \mathbb{R}^n\). We provide a few historically important examples, but refer the reader to [29] for a full treatment.

3 Unique continuation from Cauchy data↩︎

We prove 3 and 5 en route to proving 2.

3.1 Lipschitz stationary solutions↩︎

Before proving 3, we recall the interior regularity theory for Lipschitz stationary solutions to the MSS 5 , which follows from the theory for outer stationary maps (i.e. solutions to the outer variation system). Assume only that \(u: \Omega\subset \mathbb{R}^k \rightarrow \mathbb{R}^{n-k}\) is outer stationary and \(u \in C_{\text{loc}}^1(\Omega; \mathbb{R}^{n-k})\). Fix \(\mathbf{x}_0 \in \Omega\) and choose \(r > 0\) so that \(B_r^k(\mathbf{x}_0) \subset \subset \Omega\). An important fact is that \(\mathcal{A}\) is strongly rank-one convex when restricted to Lipschitz functions [34] Lemma 6.7: the lemma is stated for \(k = 2\) but generalizes to arbitrary\(k\). Therefore, we can differentiate the equation in \(B_r^k(\mathbf{x}_0)\) to obtain a quasi-linear system with \(C^{0}\) coefficients satisfying the Legendre–Hadamard ellipticity condition [35]. The classical \(W^{2, p}\) theory for linear elliptic systems (see e.g. [35], [36]) shows that \(u \in C^{1, \alpha}(B_r^k(\mathbf{x}_0); \mathbb{R}^{n-k})\).⁴ Morrey’s Theorem [36] then gives \(u \in C^\omega(B_r^k(\mathbf{x}_0); \mathbb{R}^{n-k})\) (i.e. the analytic functions). As a result, \(C^1\) stationary solutions to 5 are analytic in the interior. In general, this cannot be improved due to the LOC. The outer variation system, 5 , and 6 are equivalent at regular points. In particular, when \(n-k = 1\) the MSS reduces to the MSE since Lipschitz solutions are analytic in the interior by the De Giorgi–Nash–Moser Theorem [37].

Since Lipschitz solutions to the MSE are analytic and solve a uniformly elliptic divergence form scalar equation, the SUCP and uniqueness in the Cauchy problem are immediate when \(n-k = 1\), provided enough boundary regularity is assumed. This is not the case for high-codimension Lipschitz stationary graphs since there exist examples which are singular (see e.g. [32], [38], [39]). Hence, the set of interior regular points for a Lipschitz stationary graph is not apriori known to be connected. This is not an issue when \(k \leq 3\), since all such stationary graphs are analytic in the interior (see e.g. [40]–[42]).

In [25], it was shown that the interior singular set of a \(k\)-dimensional Lipschitz stationary solution to 5 in \(\mathbb{R}^n\) is relatively closed in the domain \(\Omega\subset \mathbb{R}^k\) and has Hausdorff dimension at most \(k-4\). Denote this set \(\mathop{\mathrm{Sing}}(u) \subset \Omega\) for a given Lipschitz stationary solution to 5 on \(\Omega\). The following lemma is essential in our argument as it implies that \(\mathop{\mathrm{Reg}}(u) \mathrel{\vcenter{:}}= \Omega\setminus \mathop{\mathrm{Sing}}(u)\) is connected, open, and dense in \(\Omega\). An elegant proof by Mooney can be found in [43].

The final ingredient is the boundary regularity theory. When \(n-k = 1\), we can apply boundary Schauder estimates. For general \(n,k\), we need Allard’s boundary regularity theory in [10] which requires strict convexity of \(\partial\Omega\). See [32] for a proof.

The utility of 15 is that it gives us some wiggle room near \(\partial\Omega\) to set up a Cauchy problem 2 with \(L\) as in 1 along regions of \(\partial\Omega\) with enough regularity. Suppose \(u \in C_{\text{loc}}^{0}(\overline{\Omega}; \mathbb{R}^{n-k}) \cap C_{\text{loc}}^{0,1}(\Omega; \mathbb{R}^{n-k})\), \(u\) solves 5 in a domain \(\Omega\) (not necessarily bounded) with Lipschitz boundary and that \(u\) has Cauchy data ?? . Let \(v \in C_{\text{loc}}^{0}(\overline{\Omega}; \mathbb{R}^{n-k}) \cap C_{\text{loc}}^{0,1}(\Omega; \mathbb{R}^{n-k})\) be any other solution to 5 in \(\Omega\) with Cauchy data ?? .

A simple example comes from applying 3 to the LOC.

Existence and uniqueness of solutions to the Dirichlet problem for the MSS are not guaranteed, even when \(\Omega= B_1^k(0)\) and we assume \(C^\infty\) boundary data (see e.g. [32], [38]). However, we will see that 3 can be used to establish uniqueness in the Plateau problem.

3.2 Area-minimizing currents↩︎

We formulate and prove the unique continuation property from Cauchy data for area-minimizing integral currents using 3, the interior regularity theory of Almgren in [26] (see also [45], [46], [47]), and the boundary regularity theory of De Lellis, De Philippis, Hirsch, and Massaccesi in [12]. Going forward, we let \(\Gamma\) and \(T\) be as in 1 unless otherwise stated.

We will need to distinguish between interior regular points and boundary regular points for \(T\) since we would like to locally reduce the problem to the case of Lipschitz stationary graphs using the regularity theory for area-minimizing integral currents. Before proving 5, we summarize the key points from the regularity theory beginning with the definition of interior regular points.

When combined with a covering argument, analyticity yields the following identity lemma for connected interior regular sets. The requirement of a shared boundary is essential. We leave the details of the proof to the reader.

Defining boundary regular points is more delicate. The following definition depends on the boundary regularity assumed in 1.

Fix \(p \in \mathop{\mathrm{Reg}}_b(T)\) (so \(p \in \Gamma)\) and let \(\Sigma\) be as in 20. Choose \(r >0\) so small that \(B^n_r(p) \cap \Sigma\) is diffeomorphic to a \(k\)-dimensional disk. Then the Constancy Theorem implies:

1. \(\Gamma\cap B^n_r(p) \subset \Sigma\) and divides \(\Sigma\) into two disjoint \(k\)-dimensional oriented connected embedded \(C^{3, \alpha}\) submanifolds \(\Sigma^{\pm}\) with \(\partial\Sigma^\pm = \pm \Gamma\).

2. There is a natural number \(Q \in \mathbb{N}\) such that \[T \llcorner B^n_r(p) = Q[[\Sigma^+]] + (Q - 1)[[\Sigma^-]].\]

The number \(Q\) is the multiplicity of \(T\) at \(p \in \mathop{\mathrm{Reg}}_b(T)\). The density of \(T\) at \(p \in \mathop{\mathrm{Reg}}_b(T)\) is \[\Theta_T(p) := Q - \frac{1}{2},\] and coincides with the definition 9 by [12]. The points \(p\) at which \(Q = 1\) are called density \(\frac{1}{2}\) points, or one-sided points. The term “one-sided" comes from the fact that \(T \llcorner B^n_r(p) = [[\Sigma^+]]\). In other words, \(T\) can be locally identified with \(\Sigma^+\) near \(p\). If \(Q > 1\), we say that \(p\) is a two-sided point. See [12] for a helpful illustration of one-sided and two-sided boundary regular points. Since we intend to utilize PDE techniques, we need conditions under which a relatively open subset of one-sided regular points exists along \(\Gamma\).

The question of the existence of one-sided boundary regular points for suitably regular \(\Gamma\) dates back to Almgren in his Big Regularity Paper [26]. A classic result of Hardt and Simon [27] says that if \(k = n - 1\), then \(\mathop{\mathrm{Reg}}_b(T) = \Gamma\). However, without additional hypotheses we cannot be sure that a given point \(p \in \mathop{\mathrm{Reg}}_b(T)\) is one-sided for general \(1 \leq k \leq n-1\). The first positive results on the existence of one-sided regular points in the high-codimension case were proved recently by De Lellis, De Philippis, Hirsch, and Massaccesi [12]. We record the most relevant.

We can now prove 5.

4 Uniqueness of compactly supported calibrated currents↩︎

We are ready to prove 2. This is accomplished by showing that the hypotheses of 5 are met for any such area-minimizing currents. We start with two general lemmas that are only related to calibrations. Suppose \(e_1,\ldots, e_n\) is an orthonormal basis for \(\mathbb{R}^n\). Let \(1 \leq k \leq n\). For each \(i = 1,\ldots, k\), we define \[e_i \, \lrcorner \, (e_1 \wedge \cdots \wedge e_k) \mathrel{\vcenter{:}}= (-1)^{i-1} e_1 \wedge \cdots \wedge\widehat{e_i} \wedge \cdots \wedge e_k.\] Hence, for each \(i\) we have \[e_i \wedge(e_i \, \lrcorner \, (e_1 \wedge \cdots \wedge e_k)) = e_1 \wedge\cdots \wedge e_k.\] The following lemma has been adapted from [49], and is a consequence of the fact that a comass one \(k\)-form \(\varphi\in \mathcal{E}^k(\mathbb{R}^n)\) achieves its pointwise maximum on its contact set. Since the proof is standard, we omit the details.

For fixed \(i,j\), we call \(\xi_{ij}\) a first cousin of \(\xi\). The next two lemmata allow us to locally set up a Cauchy problem for the MSS if it is known that a calibrated current exists for a given \(C^{3,\alpha}\) boundary.

Given a calibrated integral current with compact support, all the area-minimizing compactly supported integral currents which have the same boundary must also be calibrated by the same form. This is proved using 12 and 9.

2 is now an immediate consequence of 27 below.

In the special case \(U = \mathbb{R}^n\), we simply choose \(W \subset \subset \mathbb{R}^n\) satisfying \(\mathop{\mathrm{spt}}T \cup \mathop{\mathrm{spt}}T^\prime \subset W\) to obtain 2. As a result, we have proved uniqueness in the oriented Plateau problem for many important classes of area-minimizing submanifolds (e.g. the examples in Section 2). While existence of solutions to the oriented Plateau problem is well understood, it is not clear when a calibrated current spans a given boundary. Thus, for now we must apply 2 case by case to determine uniqueness. Notice that 27 can be used to rule out the existence of calibrated currents spanning a fixed boundary.

5 General ambient manifolds↩︎

One can define calibrations on a general Riemannian manifold. A Riemannian manifold together with a calibration is called a calibrated manifold. There are many examples of calibrated manifolds (see e.g. [29]). However, in contrast to 13, a calibrated current in a calibrated manifold can only be assumed to be homologically area-minimizing (cf. [29]). Nonetheless, all the results in Section 3.2–4 and Appendix A carry over to a general ambient manifold \(M\) that is complete without boundary, analytic, and satisfies \(H_k(M; \mathbb{R}) = 0\).

The last two conditions are essential. For analyticity, our arguments heavily rely on the fact that \(C^1\) minimal submanifolds in an analytic ambient manifold are actually analytic by the bootstrapping argument (see also 18), so that 19 holds. For the homological condition, a general ambient manifold \(M\) may contain a closed \(k\)-dimensional submanifold \(\Sigma\) calibrated by a \(k\)-form \(\varphi\). For example, \(\Sigma= \mathbb{S}^n\) is a closed special Lagrangian submanifold in \(T^*\mathbb{S}^n\) with the Stenzel metric (see e.g. [53] or [54]). By chopping \(\Sigma\) into two submanifolds in a suitable way, it gives counterexamples to 13 and shows the non-uniqueness in the oriented Plateau problem. However, by assuming \(H_k(M; \mathbb{R}) = 0\), statements similar to 13 and 26 hold. The main reason is that we rule out the existence of closed calibrated \(k\)-dimensional submanifolds in \(M\) by the universal coefficient theorem and de Rham’s theorem.

6 Uniqueness of restrictions of global area-minimizers↩︎

In this section, we show that if \(T \in \mathcal{I}_{k, c}(\mathbb{R}^n)\) is a global area-minimizer arising as the restriction of another global area-minimizer \(T^\prime \in \mathcal{I}_{k, c}(\mathbb{R}^n)\), then \(T\) is the unique global area-minimizer among all compactly supported integral currents with the same boundary as \(T\). The authors believe that it must be known to experts in the field (particularly when stated as 32). Due to the lack of a precise reference, the proof is included for completeness.

Let \(C \subset \mathbb{R}^n\) be a \(k\)-dimensional cone (e.g. Simons’ cone or the LOC). Then \(C\) is called regular if \(C \setminus \{p\} = \mathop{\mathrm{Reg}}_i(C)\) for some \(p \in \mathbb{R}^n\). For such a \(p\), we call \(C \cap \mathbb{S}^{n-1}(p)\) its link. We have the following corollary for area-minimizing regular cones:

32 proves the corresponding claim in the fourth paragraph of Section 7 in [39].

References↩︎