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

TL;DR

This paper derives a Simons' type formula for constant mean curvature surfaces in certain homogeneous 3-manifolds and applies it to establish rigidity results under specific curvature conditions.

Contribution

It introduces a new Simons' type formula for cmc surfaces in homogeneous 3-manifolds with non-zero bundle curvature, advancing understanding of their geometric properties.

Findings

Derived a Simons' type formula for cmc surfaces in $E(, au)$

Established a rigidity theorem for cmc surfaces when  > 4^2 under a pinching condition

Provided new insights into the geometry of cmc surfaces in homogeneous 3-manifolds

Abstract

In this paper, we give a Simons' type formula for the cmc surfaces in homogeneous $3$-manifolds $E(\kappa,\tau)$, $\tau\neq0$. 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.