Willmore surfaces and Hopf tori in homogeneous 3-manifolds

Abstract

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