Calculus of Variations and Geometric Measure Theory

M. Caselli - E. Florit-Simon - J. Serra

Yau's conjecture for nonlocal minimal surfaces

created by caselli on 13 Jun 2023
modified on 01 Feb 2024



Inserted: 13 jun 2023
Last Updated: 1 feb 2024

Year: 2023

ArXiv: 2306.07100 PDF


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.