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On rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space

Weimin Sheng, Yinhang Wang, Jie Wu

Abstract

In this paper, we first give some new characterizations of geodesic spheres in the hyperbolic space by the condition that hypersurface has constant weighted shifted mean curvatures, or constant weighted shifted mean curvature ratio, which generalize the result of Hu-Wei-Zhou \cite{HWZ23}. Secondly, we investigate several rigidity problems for hypersurfaces in the hyperbolic space with constant linear combinations of weighted shifted mean curvatures as well as radially symmetric shifted mean curvatures. As applications, we obtain the rigidity results for hypersurfaces with constant linear combinations of mean curvatures in a general form and constant Gauss-Bonnet curvature $L_k$ under weaker conditions, which extend the work of the third author and Xia \cite{WX14}.

On rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space

Abstract

In this paper, we first give some new characterizations of geodesic spheres in the hyperbolic space by the condition that hypersurface has constant weighted shifted mean curvatures, or constant weighted shifted mean curvature ratio, which generalize the result of Hu-Wei-Zhou \cite{HWZ23}. Secondly, we investigate several rigidity problems for hypersurfaces in the hyperbolic space with constant linear combinations of weighted shifted mean curvatures as well as radially symmetric shifted mean curvatures. As applications, we obtain the rigidity results for hypersurfaces with constant linear combinations of mean curvatures in a general form and constant Gauss-Bonnet curvature under weaker conditions, which extend the work of the third author and Xia \cite{WX14}.
Paper Structure (6 sections, 20 theorems, 102 equations)

This paper contains 6 sections, 20 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm1.1']}
  4. Proof of Theorem \ref{['thm1.3']}
  5. Proof of Theorems \ref{['thm+1']}--\ref{['thm+3']}
  6. Proof of Corollaries \ref{['coro1']} and \ref{['coro2']}

Key Result

Theorem A

Let $\Sigma$ be a closed embedded hypersurface in $\mathbb{H}^{n+1}$. If either of the following conditions holds on $\Sigma$: then $\Sigma$ is a centered geodesic sphere.

Theorems & Definitions (35)

  • Theorem A: W16
  • Theorem B: HWZ23
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.1
  • Remark 1.2
  • ...and 25 more