Preprint
Inserted: 13 jun 2023
Last Updated: 1 feb 2024
Year: 2023
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.
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