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

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 $\mathrm{\mid }g{\mathrm{\mid }}_{C^{1,\alpha }}\le \mathrm{\Gamma }$, we show that there exists a constant $C$ depending only on $n,h,A,\mathrm{\Gamma }$ and geometric quantities such that: $$\sum _{i=0}^n{b}^i(V)\le C(1+\text{index}(V))\text{if }3\le n+1\le 7;$$ $$M^{\ast n-7}(\text{sing}(V))\le C(1+\text{index}(V))\text{if }n+1\ge 8.$$ Here, $b^i$ denote the Betti numbers over any field, $M^{\ast n-7}$ 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.