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

TL;DR

This paper demonstrates the existence of infinitely many hypersurfaces with prescribed mean curvature in certain manifolds containing a strictly stable minimal surface, extending previous constructions to a broader class of functions.

Contribution

It introduces a novel method combining multiple techniques to construct infinitely many prescribed mean curvature hypersurfaces in manifolds with specific stability properties.

Findings

Infinitely many distinct PMC hypersurfaces exist under given conditions.

Construction applies to manifolds with nontrivial second homology or lacking the Frankel property.

Method synthesizes ideas from prior minimal and constant mean curvature surface constructions.

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.