Quantitative Estimates on the Topology and Singular Set of Prescribed Mean Curvature Hypersurfaces

TL;DR

This paper provides quantitative bounds on the topology and singular set of prescribed mean curvature hypersurfaces in Riemannian manifolds, extending previous results from minimal to PMC hypersurfaces and applying to recent min-max constructions.

Contribution

It extends topological and singularity estimates from minimal to prescribed mean curvature hypersurfaces, incorporating bounds on the mean curvature function and applying to recent min-max examples.

Findings

Betti number bounds in terms of index for PMC hypersurfaces

Minkowski content bounds on singular sets for high dimensions

Extension of previous minimal hypersurface results to PMC setting

Abstract

We establish quantitative topological and singularity properties for (certain) prescribed mean curvature (PMC) hypersurfaces $V^n$ in Riemannian manifolds $(N^{n+1},h)$. Indeed, if $V$ has area at most $A>0$ with PMC given by a $C^{1,\alpha}$ function $g:N\to \mathbb{R}$ with the bound $|g|_{C^{1,\alpha}}\leq \Gamma$, we show that there exists a constant $C$ depending only on $n,h,A,\Gamma$ and geometric quantities such that: \[\sum^n_{i=0}b^i(V) \leq C(1+\text{index}(V)) \quad \text{if }3\leq n+1\leq 7;\] \[M^{*n-7}(\text{sing}(V)) \leq C(1+\text{index}(V)) \quad \text{if }n+1\geq 8.\] Here, $b^i$ denote the Betti numbers over any field, $M^{*n-7}$ denotes the upper $(n-7)$-dimensional Minkowski content, and $\text{sing}(V)$ is the singular set of $V$. The first inequality extends the work of Song from the minimal hypersurface setting to the PMC hypersurface setting, whilst the second extends work of the authors. Our results apply to the PMC hypersurfaces constructed recently through min-max techniques by Bellettini--Wickramasekera.