2023/08/11 by Hardy Chan, Serena Dipierro, Chan, Hardy +5 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2308.06328
openalex publication_date 2023/08/11 · openalex created_date 2023/08/16 · openalex updated_date 2026/08/02
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 ∂ E is a stable s-minimal surface in B1⊂ \mathbb R3 then: - ∂ E∩ B1/2 enjoys a C2,α estimate that is robust as s\uparrow 1 (i.e. uniform in s); - the distance between different connected components of~∂ E∩ B1/2 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 R4, for s∈(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).