Yau’s conjecture for nonlocal 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 (without any genericity assumption on the metric), 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 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” and the compactness of the class of finite index $s$-minimal surfaces in the strongest geometric sense (that is, they are shown to subsequentially converge smoothly and with multiplicity one). These properties make nonlocal minimal surfaces ideal objects on which to apply min-max variational methods as well as to approximate classical minimal surfaces. Combined with recent results by Chan–Dipierro–Serra–Valdinoci (2023) and Florit-Simon (2024), which show the convergence (as $s\uparrow 1$) of the nonlocal $s$-minimal surfaces constructed here to smooth, classical minimal surfaces in three dimensions, this work sets a new, powerful method for the study of old and new questions on the existence of classical minimal surfaces.