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