A Simons' type formula for cmc surfaces in homogeneous $3$-manifolds

Abstract

In this paper, we give a Simons' type formula for the cmc surfaces in homogeneous $3$-manifolds $E(\kappa ,\tau )$, $\tau \ne 0$. As an application, we give a rigidity result in the case of $\kappa >4{\tau }^2$ for the cmc surfaces under a pinching assumption of the second fundamental form.

9 pages