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Sharp Minkowski Type Inequality in Cartan-Hadamard 3-Spaces

Fang Hong

Abstract

In this paper, we proved a sharp Minkowski type inequality in Cartan-Hadamard 3-spaces by harmonic mean curvature flow and improves the known estimates for total mean curvature in hyperbolic 3-space. In particular, we sharpened Ghomi-Spruck's result. As a corollary, we also get a comparison theorem between total mean curvature in Cartan-Hadamard 3-spaces with that of the geodesic sphere in hyperbolic 3-space with constant curvature.

Sharp Minkowski Type Inequality in Cartan-Hadamard 3-Spaces

Abstract

In this paper, we proved a sharp Minkowski type inequality in Cartan-Hadamard 3-spaces by harmonic mean curvature flow and improves the known estimates for total mean curvature in hyperbolic 3-space. In particular, we sharpened Ghomi-Spruck's result. As a corollary, we also get a comparison theorem between total mean curvature in Cartan-Hadamard 3-spaces with that of the geodesic sphere in hyperbolic 3-space with constant curvature.
Paper Structure (7 sections, 8 theorems, 59 equations)

This paper contains 7 sections, 8 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations of Convexity
  4. Notations and Facts about Profile Functions
  5. Notations and Facts of Geometric Flows
  6. Proof of Theorem \ref{['thm:minkowski1']} and Corollary \ref{['cor:minkowski1']}
  7. Comparing Inequality \ref{['eq:minkowski1']} to \ref{['eq:minkowski3']}

Key Result

Theorem 1.1

For any strictly convex surface $\Gamma$ embedded in Euclidean space $\mathbb{R}^3$, where $S(\Gamma)$ denotes the surface area of $\Gamma$, $M(\Gamma) := \int_\Gamma Hd\mu$ is defined to be total mean curvature of $\Gamma$, in which the mean curvature of $\Gamma$ is given by the trace of second fundamental form $H:=\textup{trace}(\mathrm{I\!I}_\Gamma)$. Equality holds only when $\G

Theorems & Definitions (11)

  • Theorem 1.1: H. Minkowski, 1903
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Theorem 2.2: B. Kleiner, 1992
  • Proposition 3.1
  • proof : Proof of Theorem \ref{['thm:minkowski1']} and Corollary \ref{['cor:minkowski1']}
  • Theorem 3.2
  • Proposition 4.1
  • ...and 1 more