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Weyl Law and convergence in the classical limit for min-max nonlocal minimal surfaces

Enric Florit-Simon

TL;DR

The article develops a nonlocal min-max framework for $s$-minimal hypersurfaces on closed manifolds, proving a Weyl-type law for their fractional perimeters and establishing a robust compactness theory as $s\to1$ in dimension $3$ so that min-max $s$-minimal surfaces converge to smooth classical minimal surfaces. The Weyl Law is first proven for a distance-kernel energy and then transferred to the canonical fractional perimeter, enabling a uniform comparison and a limit law $\lim_{\mathfrak p\to\infty} \mathfrak p^{-1/n} l_1(\mathfrak p,M)$ that mirrors known results for classical minimal surfaces. The compactness theory combines a first/second variation analysis with uniform BV-type estimates and almost-stability to obtain $L^{-2}$ separation and uniform $C^{2,\alpha}$ control in dimension $3$, culminating in the convergence of $s$-minimizing surfaces with bounded index to smooth classical minimal surfaces via varifold convergence. These results recover density and equidistribution statements for minimal surfaces in generic metrics, illustrating that nonlocal minimal surfaces form a clean, scale-invariant regularization of the area functional and a competitive alternative to established min-max approaches. The framework thus ties nonlocal approximations to classical minimal surface theory and supports a deeper Yau-type program in the nonlocal setting.

Abstract

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 $s\in(0,1)$. 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 $s\to 1$) 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.

Weyl Law and convergence in the classical limit for min-max nonlocal minimal surfaces

TL;DR

The article develops a nonlocal min-max framework for -minimal hypersurfaces on closed manifolds, proving a Weyl-type law for their fractional perimeters and establishing a robust compactness theory as in dimension so that min-max -minimal surfaces converge to smooth classical minimal surfaces. The Weyl Law is first proven for a distance-kernel energy and then transferred to the canonical fractional perimeter, enabling a uniform comparison and a limit law that mirrors known results for classical minimal surfaces. The compactness theory combines a first/second variation analysis with uniform BV-type estimates and almost-stability to obtain separation and uniform control in dimension , culminating in the convergence of -minimizing surfaces with bounded index to smooth classical minimal surfaces via varifold convergence. These results recover density and equidistribution statements for minimal surfaces in generic metrics, illustrating that nonlocal minimal surfaces form a clean, scale-invariant regularization of the area functional and a competitive alternative to established min-max approaches. The framework thus ties nonlocal approximations to classical minimal surface theory and supports a deeper Yau-type program in the nonlocal setting.

Abstract

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 -minimal hypersurfaces (via min-max methods) on any closed -dimensional Riemannian manifold , establishing a full analogue of Yau's conjecture for . 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 ) for min-max -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.
Paper Structure (30 sections, 62 theorems, 443 equations)

This paper contains 30 sections, 62 theorems, 443 equations.

Table of Contents

  1. Introduction and main results
  2. Overview of the article and further directions
  3. Setting
  4. The fractional setting on a manifold
  5. Definitions
  6. Fundamental properties
  7. Monotonicity formula
  8. Min-max procedure -- The nonlocal volume spectrum
  9. The Weyl Law
  10. The Weyl Law with the distance kernel
  11. The distance kernel
  12. Euclidean domain case
  13. Closed manifold case -- Proof of Proposition \ref{['prop:weyld']}
  14. Equivalence with the canonical Sobolev kernel -- Proof of Theorem \ref{['thmweyl']}
  15. The Weyl Law in the classical limit $s\to 1$ -- Proof of Theorem \ref{['clasweylthm']}
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $(M^n, g)$ be an $n$-dimensional, closed Riemannian manifold. Then, for every $s\in(0,1)$ and every natural number $\mathfrak p\in\mathbb{N}^+$, there exists an $s$-minimal surface $E_{\mathfrak p}^s$ with Morse index at most $\mathfrak p$ (in the sense of Definition WeakMorseDef) and fractional where $c(M)$ and $C(M)$ are uniform as $s\to 1$. In particular, $M$ contains infinitely many $s$-mi

Theorems & Definitions (136)

  • Theorem 1.1: Fractional Yau Conjecture CFS
  • Theorem 1.2: Weyl Law for nonlocal minimal surfaces
  • Theorem 1.3: Weyl Law in the classical limit
  • Theorem 1.4: Compactness of $s$-minimal surfaces
  • Corollary 1.5: Attainability of min-max widths
  • Theorem 1.6: Density
  • Theorem 1.7: Equidistribution
  • Definition 3.1: $s$-minimal surface
  • Definition 3.2
  • Definition 3.3: Morse index and stability
  • ...and 126 more