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New Heintze-Karcher type inequalities in sub-static warped product manifolds

Haizhong Li, Yong Wei, Botong Xu

TL;DR

This work extends sharp Heintze-Karcher-type inequalities to curved ambient spaces by incorporating shifted principal curvatures. It develops two main results: a HK-type inequality for non mean-convex domains in $\mathbb{H}^{n+1}$ and a general shifted HK inequality in sub-static warped product manifolds with shift $\varepsilon$, under static-convexity. The proofs combine unit normal flow with Minkowski-type formulas and derive rigidity statements: equality forces umbilic geodesic spheres or slices, leading to Alexandrov-type theorems for a class of curvature equations. These results broaden the toolkit for geometric and isoperimetric-type inequalities in hyperbolic and warped-product settings and have applications to curvature problems and uniqueness questions.

Abstract

In this paper, we prove Heintze-Karcher type inequalities involving the shifted mean curvature for smooth bounded domains in certain sub-static warped product manifolds. In particular, we prove a Heintze-Karcher-type inequality for non mean-convex domains in the hyperbolic space. As applications, we obtain uniqueness results for hypersurfaces satisfying a class of curvature equations.

New Heintze-Karcher type inequalities in sub-static warped product manifolds

TL;DR

This work extends sharp Heintze-Karcher-type inequalities to curved ambient spaces by incorporating shifted principal curvatures. It develops two main results: a HK-type inequality for non mean-convex domains in and a general shifted HK inequality in sub-static warped product manifolds with shift , under static-convexity. The proofs combine unit normal flow with Minkowski-type formulas and derive rigidity statements: equality forces umbilic geodesic spheres or slices, leading to Alexandrov-type theorems for a class of curvature equations. These results broaden the toolkit for geometric and isoperimetric-type inequalities in hyperbolic and warped-product settings and have applications to curvature problems and uniqueness questions.

Abstract

In this paper, we prove Heintze-Karcher type inequalities involving the shifted mean curvature for smooth bounded domains in certain sub-static warped product manifolds. In particular, we prove a Heintze-Karcher-type inequality for non mean-convex domains in the hyperbolic space. As applications, we obtain uniqueness results for hypersurfaces satisfying a class of curvature equations.

Paper Structure

This paper contains 10 sections, 15 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Heintze-Karcher type inequalities for non mean-convex domains in $\mathbb{H}^{n+1}$
  3. Heintze-Karcher type inequalities in sub-static warped product manifolds
  4. Preliminaries
  5. Elementary symmetric polynomials
  6. Hypersurfaces in warped product manifolds
  7. Hypersurfaces in space forms
  8. Proof of Theorem \ref{['thm-+1 HK ineq']}
  9. Proof of Theorem \ref{['thm-shifted H-K ineq']}
  10. Proofs of Theorem \ref{['thm-+1 shifted-eq']} and Theorem \ref{['thm-s1:shifted-eq']}

Key Result

Theorem 1.1

Let $\Omega$ be a bounded domain with smooth boundary $\Sigma = \partial \Omega$ in $\mathbb{H}^{n+1}$. Assume that the mean curvature $p_1(\kappa)$ of $\Sigma$ satisfies $p_1 (\kappa) >-1$. Then Equality holds in s1:HK-main-eps=-1 if and only if $\Sigma$ is umbilic, and thus $\Sigma$ is a geodesic sphere.

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • ...and 17 more