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arXiv · 2306.07100

Yau's conjecture for nonlocal minimal surfaces

Michele Caselli·Enric Florit-Simon·Joaquim Serra
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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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Michele Caselli, Enric Florit-Simon, Joaquim Serra. 2023-06-12. Yau's conjecture for nonlocal minimal surfaces. https://arxiv.org/abs/2306.07100

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