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Willmore surfaces and Hopf tori in homogeneous 3-manifolds

Alma L. Albujer, Fábio R. dos Santos

Abstract

Some classification results for closed surfaces in Berger spheres are presented. On the one hand, a Willmore functional for isometrically immersed surfaces into an homogeneous space $\mathbb{E}^{3}(κ,τ)$ with isometry group of dimension $4$ is defined and its first variational formula is computed. Then, we characterize Clifford and Hopf tori as the only Willmore surfaces satifying a sharp Simons-type integral inequality. On the other hand, we also obtain some integral inequalities for closed surfaces with constant extrinsic curvature in $\mathbb{E}^3(κ,τ)$, becoming equalities if and only if the surface is a Hopf torus in a Berger sphere.

Willmore surfaces and Hopf tori in homogeneous 3-manifolds

Abstract

Some classification results for closed surfaces in Berger spheres are presented. On the one hand, a Willmore functional for isometrically immersed surfaces into an homogeneous space with isometry group of dimension is defined and its first variational formula is computed. Then, we characterize Clifford and Hopf tori as the only Willmore surfaces satifying a sharp Simons-type integral inequality. On the other hand, we also obtain some integral inequalities for closed surfaces with constant extrinsic curvature in , becoming equalities if and only if the surface is a Hopf torus in a Berger sphere.
Paper Structure (5 sections, 12 theorems, 131 equations)

This paper contains 5 sections, 12 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Simons-type formula for the Cheng-Yau operator in $\mathbb{E}^{3}(\kappa,\tau)$
  4. Willmore surfaces in $\mathbb{S}_b^3(\kappa,\tau)$
  5. Classification results for constant extrinsic curvature closed surfaces

Key Result

Lemma 2.1

Dillen:02 Let $\Sigma^{2}$ be an isometrically immersed parallel surface into the homogeneous space $\mathbb{E}^{3}(\kappa,\tau)$, $\kappa-4\tau^2\neq 0$. Then,

Theorems & Definitions (25)

  • Lemma 2.1
  • Remark 3.1
  • Proposition 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Proposition 4.1
  • proof
  • Remark 4.2
  • Lemma 4.3
  • ...and 15 more