Nonlocal approximation of minimal surfaces: optimal estimates from stability
Hardy Chan, Serena Dipierro, Joaquim Serra, Enrico Valdinoci
TL;DR
The paper develops a nonlocal approach to approximating minimal surfaces by studying stable $s$-minimal hypersurfaces in $\mathbb{R}^n$ as $s\uparrow 1$. It proves robust $C^{2,\alpha}$ regularity and optimal intersheet separation for stable $s$-minimal surfaces in $\mathbb{R}^3$, and derives a limit Toda-type system describing inter-sheet interactions at scale $\sqrt{1-s}$; it also classifies stable $s$-minimal cones in $\mathbb{R}^4$ for $s$ near $1$, showing hyperplanes are the only possibilities. The results illuminate the structure of nonlocal minimal surfaces, enabling a rigorous nonlocal approximation of the classical area functional and contributing to the broader program of nonlocal geometric analysis. The combination of robust curvature control, a Toda-type interaction framework, and the cone classification provides foundational tools for future nonlocal min-max constructions and potential higher-codimension extensions.
Abstract
Minimal surfaces in closed 3-manifolds are classically constructed via the Almgren-Pitts approach. The Allen-Cahn approximation has proved to be a powerful alternative, and Chodosh and Mantoulidis (in Ann. Math. 2020) used it to give a new proof of Yau's conjecture for generic metrics and establish the multiplicity one conjecture. The primary goal of this paper is to set the ground for a new approximation based on nonlocal minimal surfaces. More precisely, we prove that if $\partial E$ is a stable $s$-minimal surface in $B_1\subset \mathbb R^3$ then: - $\partial E\cap B_{1/2}$ enjoys a $C^{2,α}$ estimate that is robust as $s\uparrow 1$ (i.e. uniform in $s$); - the distance between different connected components of~$\partial E\cap B_{1/2}$ must be at least of order~$(1-s)^{\frac 1 2}$ (optimal sheet separation estimate); - interactions between multiple sheets at distances of order $(1-s)^{\frac 1 2}$ are described by the Dávila--del Pino--Wei system. A second important goal of the paper is to establish that hyperplanes are the only stable $s$-minimal hypersurfaces in $\mathbb R^4$, for $s\in(0,1)$ sufficiently close to $1$. This is done by exploiting suitable modifications of the results described above. In this application, it is crucially used that our curvature and separations estimates hold without any assumption on area bounds (in contrast to the analogous estimates for Allen-Cahn).