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 isometrically immersed 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 computed. Then, we characterize Clifford Hopf tori as only satisfying sharp Simons-type integral inequality. other also obtain some inequalities constant extrinsic curvature {E}}^3(\kappa , becoming equalities if surface torus sphere.