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Plateau's problem via the Allen--Cahn functional

Marco A. M. Guaraco, Stephen Lynch

Abstract

Let $Γ$ be a compact codimension-two submanifold of $\mathbb{R}^n$, and let $L$ be a nontrivial real line bundle over $X = \mathbb{R}^n \setminus Γ$. We study the Allen--Cahn functional, \[E_\varepsilon(u) = \int_X \varepsilon \frac{|\nabla u|^2}{2} + \frac{(1-|u|^2)^2}{4\varepsilon}\,dx,\] on the space of sections $u$ of $L$. Specifically, we are interested in critical sections for this functional and their relation to minimal hypersurfaces with boundary equal to $Γ$. We first show that, for a family of critical sections with uniformly bounded energy, in the limit as $\varepsilon \to 0$, the associated family of energy measures converges to an integer rectifiable $(n-1)$-varifold $V$. Moreover, $V$ is stationary with respect to any variation which leaves $Γ$ fixed. Away from $Γ$, this follows from work of Hutchinson--Tonegawa; our result extends their interior theory up to the boundary $Γ$. Under additional hypotheses, we can say more about $V$. When $V$ arises as a limit of critical sections with uniformly bounded Morse index, $Σ:= \operatorname{supp} \|V\|$ is a minimal hypersurface, smooth away from $Γ$ and a singular set of Hausdorff dimension at most $n-8$. If the sections are globally energy minimizing and $n = 3$, then $Σ$ is a smooth surface with boundary, $\partial Σ= Γ$ (at least if $L$ is chosen correctly), and $Σ$ has least area among all surfaces with these properties. We thus obtain a new proof (originally suggested in a paper of Fröhlich and Struwe) that the smooth version of Plateau's problem admits a solution for every boundary curve in $\mathbb{R}^3$. This also works if $4 \leq n\leq 7$ and $Γ$ is assumed to lie in a strictly convex hypersurface.

Plateau's problem via the Allen--Cahn functional

Abstract

Let be a compact codimension-two submanifold of , and let be a nontrivial real line bundle over . We study the Allen--Cahn functional, on the space of sections of . Specifically, we are interested in critical sections for this functional and their relation to minimal hypersurfaces with boundary equal to . We first show that, for a family of critical sections with uniformly bounded energy, in the limit as , the associated family of energy measures converges to an integer rectifiable -varifold . Moreover, is stationary with respect to any variation which leaves fixed. Away from , this follows from work of Hutchinson--Tonegawa; our result extends their interior theory up to the boundary . Under additional hypotheses, we can say more about . When arises as a limit of critical sections with uniformly bounded Morse index, is a minimal hypersurface, smooth away from and a singular set of Hausdorff dimension at most . If the sections are globally energy minimizing and , then is a smooth surface with boundary, (at least if is chosen correctly), and has least area among all surfaces with these properties. We thus obtain a new proof (originally suggested in a paper of Fröhlich and Struwe) that the smooth version of Plateau's problem admits a solution for every boundary curve in . This also works if and is assumed to lie in a strictly convex hypersurface.
Paper Structure (21 sections, 26 theorems, 243 equations)

This paper contains 21 sections, 26 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. Minimizing sections and Plateau's problem
  3. Key steps in the proofs
  4. An example
  5. Acknowledgements
  6. Real line bundles
  7. Real line bundles and $\operatorname{Hom}(H_1(M),\mathop{\mathrm{\mathbb{Z}}}\nolimits_2)\simeq H^1(M,\mathop{\mathrm{\mathbb{Z}}}\nolimits_2)$
  8. Real line bundles from classes of loops mod $2$
  9. Real line bundles from classes of hypersurfaces mod $2$
  10. A priori estimates for critical sections
  11. Derivative estimates at the boundary
  12. Energy monotonicity
  13. Interior estimates
  14. $L^1$-convergence of the discrepancy
  15. Stationary varifolds from critical sections
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $B \subset \mathbb{R}^n$ be an open ball and set $X = B \setminus \Gamma$. We assume $X$ is diffeomorphic to the complement of an $(n-2)$-plane in $\mathbb{R}^n$, and denote by $L$ the nontrivial real line bundle over $X$. Fix a sequence $\varepsilon_k \to 0$, and for each $k$ suppose $u_k$ is a Possibly after passing to a subsequence, we have the following behaviour:

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 2.1
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • ...and 40 more