Willmore surfaces and Hopf tori in homogeneous 3-manifolds

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.