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

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 $H_n(M,{\mathbb{Z}}_2)\ne 0$, 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.

40 pages, 14 figures. Comments welcome! V2 Updated references. V3 slight improvement of main theorems, minor remarks added. arXiv admin note: text overlap with arXiv:2011.04136, arXiv:1806.08816, arXiv:1901.01173 by other authors