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Total mean curvature surfaces in the product space $\mathbb{S}^n\times\mathbb{R}$ and applications

Alma L. Albujer, Sylvia F. da Silva, Fábio R. dos Santos

Abstract

The total mean curvature functional for submanifolds into the Riemannian product space $\mathbb{S}^n\times\mathbb{R}$ is considered and its first variational formula is presented. Later on, two second order differential operators are defined and a nice integral inequality relating both of them is proved. Finally we prove our main result: an integral inequality for closed stationary $\mathcal{H}$-surfaces in $\mathbb{S}^n\times\mathbb{R}$, characterizing the cases where the equality is attained.

Total mean curvature surfaces in the product space $\mathbb{S}^n\times\mathbb{R}$ and applications

Abstract

The total mean curvature functional for submanifolds into the Riemannian product space is considered and its first variational formula is presented. Later on, two second order differential operators are defined and a nice integral inequality relating both of them is proved. Finally we prove our main result: an integral inequality for closed stationary -surfaces in , characterizing the cases where the equality is attained.
Paper Structure (5 sections, 109 equations)

This paper contains 5 sections, 109 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The first variation of the total mean curvature
  4. Two key lemmas
  5. Proof of Theorem \ref{['teo:1']}

Theorems & Definitions (7)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof : Proof of Theorem \ref{['teo:1']}