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

Nicolau S. Aiex, Sean McCurdy, Paul Minter

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 $C^{1,α}$ function $g:N\to \mathbb{R}$ with the bound $|g|_{C^{1,α}}\leq Γ$, we show that there exists a constant $C$ depending only on $n,h,A,Γ$ 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.

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

Abstract

We establish quantitative topological and singularity properties for (certain) prescribed mean curvature (PMC) hypersurfaces in Riemannian manifolds . Indeed, if has area at most with PMC given by a function with the bound , we show that there exists a constant depending only on and geometric quantities such that: Here, denote the Betti numbers over any field, denotes the upper -dimensional Minkowski content, and is the singular set of . 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.
Paper Structure (9 sections, 21 theorems, 47 equations)

This paper contains 9 sections, 21 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Prescribed Mean Curvature Varifolds
  4. Regular and Singular Sets of a Varifold
  5. Quantitative Stratification
  6. Singular Set Estimates
  7. Ambient Dimension at Most $8$
  8. $k$-connected Sets
  9. An elementary covering lemma

Key Result

Theorem A

Let $3\leq n+1\leq 7$, $\Gamma,\Lambda,\mu,\mu_1>0$, and $I\in \{0,1,2,\dotsc\}$. Let $(N^{n+1},h)$ be a closed Riemannian manifold and fix a $C^{1,\alpha}$ function $g:N\to \mathbb{R}$ with $|g|_{1,\alpha}\leq\Gamma$. Then if $V\in\mathfrak{s}_{g,\Lambda,I,\mu,\mu_1}(N)$, one has Here, $C = C(N,h,\Gamma,\Lambda,\mu,\mu_1)\in (0,\infty)$.

Theorems & Definitions (57)

  • Theorem A
  • Theorem B: cf. Theorem \ref{['main theorem n>7']} and Theorem \ref{['main theorem n=7']}
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4: bellettini-wickramasekera2019:arxiv*Definition 1.1
  • Definition 2.5: bellettini-wickramasekera2019:arxiv*Definition 6.1
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • ...and 47 more