Plateau’s problem via the Allen–Cahn functional

Let \(\Gamma \) $\mathrm{\Gamma }$ be a compact codimension-two submanifold of \({\mathbb {R}}^n\) ${\mathbb{R}}^n$ , and let L be a nontrivial real line bundle over \(X = {\mathbb {R}}^n {\setminus } \Gamma \) $X={\mathbb{R}}^n\backslash \mathrm{\Gamma }$ . We study the Allen–Cahn functional, \(\begin{aligned}E_\varepsilon (u) = \int _X \varepsilon \frac{|\nabla u|^2}{2} + \frac{(1-|u|^2)^2}{4\varepsilon }\,dx, \\\end{aligned}\) $$\begin{array}{c}\hfill E_{\epsilon }(u)={\int }_X\epsilon \frac{{|\mathrm{\nabla }u|}^2}{2}+\frac{{(1-|u|}^2{)}^2}{4\epsilon }dx,\\ \hfill \end{array}$$ 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 \(\Gamma \) $\mathrm{\Gamma }$ . We first show that, for a family of critical sections with uniformly bounded energy, in the limit as \(\varepsilon \rightarrow 0\) $\epsilon \to 0$ , the associated family of energy measures converges to an integer rectifiable \((n-1)\) $(n-1)$ -varifold V. Moreover, V is stationary with respect to any variation which leaves \(\Gamma \) $\mathrm{\Gamma }$ fixed. Away from \(\Gamma \) $\mathrm{\Gamma }$ , this follows from work of Hutchinson–Tonegawa; our result extends their interior theory up to the boundary \(\Gamma \) $\mathrm{\Gamma }$ . Under additional hypotheses, we can say more about V. When V arises as a limit of critical sections with uniformly bounded Morse index, \(\Sigma := {{\,\textrm{supp}\,}}\Vert V\Vert \) $\mathrm{\Sigma }:=\text{supp}\Vert V\Vert$ is a minimal hypersurface, smooth away from \(\Gamma \) $\mathrm{\Gamma }$ and a singular set of Hausdorff dimension at most \(n-8\) $n-8$ . If the sections are globally energy minimizing and \(n = 3\) $n=3$ , then \(\Sigma \) $\mathrm{\Sigma }$ is a smooth surface with boundary, \(\partial \Sigma = \Gamma \) $\partial \mathrm{\Sigma }=\mathrm{\Gamma }$ (at least if L is chosen correctly), and \(\Sigma \) $\mathrm{\Sigma }$ 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 \({\mathbb {R}}^3\) ${\mathbb{R}}^3$ . This also works if \(4 \le n\le 7\) $4\le n\le 7$ and \(\Gamma \) $\mathrm{\Gamma }$ is assumed to lie in a strictly convex hypersurface.