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arXiv · 2405.00595

Infinitely Many Half-Volume Constant Mean Curvature Hypersurfaces via Min-Max Theory

Liam Mazurowski·Xin Zhou
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Abstract

Let $(M^{n+1},g)$ be a closed Riemannian manifold of dimension $3\le n+1\le 5$. We show that, if the metric $g$ is generic or if the metric $g$ has positive Ricci curvature, then $M$ contains infinitely many geometrically distinct constant mean curvature hypersurfaces, each enclosing half the volume of $M$. As an essential part of the proof, we develop an Almgren-Pitts type min-max theory for certain non-local functionals of the general form $$\Omega \mapsto \operatorname{Area}(\partial \Omega) - \int_\Omega h + f(\operatorname{Vol}(\Omega)).$$

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Liam Mazurowski, Xin Zhou. 2024-05-01. Infinitely Many Half-Volume Constant Mean Curvature Hypersurfaces via Min-Max Theory. https://arxiv.org/abs/2405.00595

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