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A Heintze-Karcher type inequality in hyperbolic space

Yingxiang Hu, Yong Wei, Tailong Zhou

TL;DR

This work extends the Heintze–Karcher framework to hyperbolic space by proving a shifted inequality for hypersurfaces with $H>n$, namely $\int_{\Sigma} \frac{V-V_{,\nu}}{H-n} d\mu \ge \frac{n+1}{n} \int_{\Omega} V dvol$, with equality characterizing umbilicity. The proof uses the unit normal flow in $\mathbb{H}^{n+1}$ and a monotone functional $Q(t)$ tied to level-set geometry, together with Minkowski-type identities and a limiting argument. The results yield an Alexandrov-type rigidity for hypersurfaces with constant shifted $k$th mean curvature and a uniqueness statement for $h$-convex domains under curvature equations $E_k(\tilde{\kappa})=\chi(V-V_{,\nu})$, thereby advancing rigidity and uniqueness theory in horospherically convex hyperbolic geometry. Overall, the paper generalizes a classical Euclidean inequality to a hyperbolic setting and clarifies the geometric implications for shifted curvature invariants.

Abstract

In this paper, we prove a new Heintze-Karcher type inequality for shifted mean convex hypersurfaces in hyperbolic space. As applications, we prove an Alexandrov type theorem for closed embedded hypersurfaces with constant shifted $k$th mean curvature in hyperbolic space. Furthermore, a uniqueness result for $h$-convex hypersurfaces satisfying certain curvature equations is obtained.

A Heintze-Karcher type inequality in hyperbolic space

TL;DR

This work extends the Heintze–Karcher framework to hyperbolic space by proving a shifted inequality for hypersurfaces with , namely , with equality characterizing umbilicity. The proof uses the unit normal flow in and a monotone functional tied to level-set geometry, together with Minkowski-type identities and a limiting argument. The results yield an Alexandrov-type rigidity for hypersurfaces with constant shifted th mean curvature and a uniqueness statement for -convex domains under curvature equations , thereby advancing rigidity and uniqueness theory in horospherically convex hyperbolic geometry. Overall, the paper generalizes a classical Euclidean inequality to a hyperbolic setting and clarifies the geometric implications for shifted curvature invariants.

Abstract

In this paper, we prove a new Heintze-Karcher type inequality for shifted mean convex hypersurfaces in hyperbolic space. As applications, we prove an Alexandrov type theorem for closed embedded hypersurfaces with constant shifted th mean curvature in hyperbolic space. Furthermore, a uniqueness result for -convex hypersurfaces satisfying certain curvature equations is obtained.
Paper Structure (6 sections, 12 theorems, 60 equations)

This paper contains 6 sections, 12 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Elementary symmetric functions
  4. Hypersurfaces in hyperbolic space
  5. Proof of Theorem \ref{['s1:main-thm']}
  6. Proof of Theorems \ref{['s1:thm-application-1']} and \ref{['s1:thm-application-2']}

Key Result

Theorem 1.1

Let $\Omega$ be a bounded domain with smooth boundary $\Sigma=\partial \Omega$ in hyperbolic space $\mathbb H^{n+1}$. Fix a point $o\in \mathbb H^{n+1}$ and $V(x)=\cosh r(x)$, where $r(x)=d(o,x)$ is the distance to this point $o$. Assume that the mean curvature of $\Sigma=\partial\Omega$ satisfies $ where $V_{,\nu}=\langle \overline\nabla V,\nu\rangle=\langle \sinh r \partial_r,\nu \rangle$ is the

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Proposition 3.1
  • Lemma 3.2
  • ...and 11 more