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Surfaces with parallel mean curvature in warped product spaces

Fernando Manfio, Verônica Reis, Feliciano Vitório

Abstract

In this work, we obtain a geometric description of surfaces $M^2$ of arbitrary codimension in the warped product $\mathbb{R}\times_ρ\mathbb{Q}^n_ε$, with parallel mean curvature vector field in the normal connection, extending a result by Alencar-do Carmo-Tribuzy.

Surfaces with parallel mean curvature in warped product spaces

Abstract

In this work, we obtain a geometric description of surfaces of arbitrary codimension in the warped product , with parallel mean curvature vector field in the normal connection, extending a result by Alencar-do Carmo-Tribuzy.
Paper Structure (4 sections, 3 theorems, 44 equations)

This paper contains 4 sections, 3 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reduction of codimension
  4. Proof of main result

Key Result

Theorem 1.1

Let $f\colon M^2\to\mathbb{R}\times_{\rho}\mathbb{Q}^n_{\epsilon}$, $n\geq 5$, be a surface with nonzero parallel mean curvature vector field. Then, one of the following possibilities holds:

Theorems & Definitions (8)

  • Theorem 1.1
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 4.1
  • proof : Proof of Theorem \ref{['teo:main']}