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 ${\mathbb{E}}^3(\kappa ,\tau )$ 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 ${\mathbb{E}}^3(\kappa ,\tau )$, becoming equalities if and only if the surface is a Hopf torus in a Berger sphere.

21 pages

References in corpus (1)

• Critical Tori for Mean Curvature Energies in Killing Submersions

Cited by in corpus (1)

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