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Yau's conjecture for nonlocal minimal surfaces

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arxiv 2306.07100 v3 pith:ENF5ZFJV submitted 2023-06-12 math.DG math.AP

Michele Caselli , Enric Florit-Simon , Joaquim Serra This is my paper · ORCID
classification math.DGmath.AP
keywords surfacesminimalnonlocalclosedfiniteindexmorseproperties
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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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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Harmonic maps to the circle with higher dimensional singular set

    math.DG 2024-11 conditional novelty 8.0 of 10

    Singular S1-valued harmonic maps with prescribed codimension-2 singular set exist on closed manifolds, and three variational relaxations share the same renormalised interaction energy.

  2. Fractional multi-phase transitions and nonlocal minimal partitions

    math.AP 2025-06 conditional novelty 7.0 of 10

    Limits of vectorial fractional Allen-Cahn solutions are stationary nonlocal minimal partitions, and minimizing 3-partitions are smooth outside a small singular set, even under a reversed triangle inequality for s clos...

  3. Gamma-convergence of nonlocal energies for partitions

    math.AP 2025-06 accept novelty 7.0 of 10

    Fractional partition perimeters Gamma-converge as s approaches 1/2 to the local partition perimeter with surface tensions replaced by their shortest-path relaxation.

  4. Stable $s$-minimal cones in $\mathbb{R}^2$ are flat for $s \sim 0$

    math.AP 2024-12 reject novelty 7.0 of 10

    The paper claims that for small values of the fractional perimeter parameter s, the only stable s-minimal cones in R^2 are half-planes, but a key integral estimate in the proof is false.

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