2024/06/18 by Enric Florit-Simon, Florit-Simon, Enric · 2 citations
Mathematics · #Geometric Analysis and Curvature Flows #Advanced Harmonic Analysis Research #Differential Equations and Boundary Problems
paper · pdf · doi:10.48550/arxiv.2406.12162
We study nonlocal minimal surfaces as a new approximation theory for the area functional, and more specifically in the context of Yau's conjecture on the existence of minimal surfaces in closed three-dimensional manifolds. This programme offers an alternative to the Almgren--Pitts and Allen--Cahn approaches, with advantageous features both from the existence and regularity viewpoints. We build on recent work in which the author and collaborators constructed infinitely many nonlocal s-minimal hypersurfaces (via min-max methods) on any closed n-dimensional Riemannian manifold M, establishing a full analogue of Yau's conjecture for s∈(0,1). The present article first proves a Weyl-type Law for the fractional perimeters of these hypersurfaces. The rest -- and main part -- of the article is devoted to obtaining uniform estimates (in the classical limit s→ 1) for min-max s-minimal surfaces in closed three-manifolds, eventually establishing their convergence to smooth, classical minimal surfaces. We recover in particular recent results on existence, generic density and equidistribution of minimal surfaces, which are a strong form of Yau's conjecture in this setting.