Yau's conjecture for nonlocal minimal surfaces

TL;DR

This paper extends the theory of minimal surfaces to a nonlocal setting on closed manifolds, proving existence, regularity, and rigidity results for these surfaces, and demonstrating their suitability for variational methods.

Contribution

It introduces nonlocal minimal surfaces on closed manifolds, proves existence of infinitely many such surfaces, and establishes regularity and rigidity properties, including a Bernstein-type result.

Findings

Existence of infinitely many nonlocal s-minimal surfaces in closed manifolds.

Regularity results: smoothness in low dimensions, partial regularity in higher dimensions.

Rigidity properties for finite Morse index s-minimal surfaces.

Abstract

We introduce nonlocal minimal surfaces on closed manifolds and establish a far-reaching Yau-type result: in every closed, $n$-dimensional Riemannian manifold we construct infinitely many nonlocal $s$-minimal surfaces. We prove that, when $s\in (0,1)$ is sufficiently close to $1$, the constructed surfaces are smooth for $n=3$ and $n=4$, while for $n\ge 5$ they are smooth outside of a closed set of dimension $n-5$. Moreover, we prove surprisingly strong regularity and rigidity properties of finite Morse index $s$-minimal surfaces such as a "finite Morse index Bernstein-type result". These properties make nonlocal minimal surfaces ideal objects on which to apply min-max variational methods.