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arXiv · 2502.07098

Infinitely Many Surfaces with Prescribed Mean Curvature in the Presence of a Strictly Stable Minimal Surface

Pedro Gaspar·Jared Marx-Kuo
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Abstract

We construct infinitely many distinct hypersurfaces with prescribed mean curvature (PMC) for a large class of prescribing functions when $(M^{n+1}, g)$ is a closed smooth manifold containing a minimal surface that is strictly stable (or more generally, admits a contracting neighborhood). In particular, we construct infinitely many distinct PMCs when $H_n(M, \mathbb{Z}_2) \neq 0$, or if $(M, g)$ does not satisfy the Frankel property. Our construction synthesizes ideas from Song's construction of infinitely many minimal surfaces in the non-generic setting, Dey's construction of multiple constant mean curvature surfaces, and Sun--Wang--Zhou's min-max construction of free boundary PMCs.

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Pedro Gaspar, Jared Marx-Kuo. 2025-02-10. Infinitely Many Surfaces with Prescribed Mean Curvature in the Presence of a Strictly Stable Minimal Surface. https://arxiv.org/abs/2502.07098

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