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 E3(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 satisfying a sharp Simons-type integral inequality. On the other hand, we also obtain some integral inequalities for closed surfaces with constant extrinsic curvature in E3(κ,τ), becoming equalities if and only if the surface is a Hopf torus in a Berger sphere.