The Abresch-Rosenberg Shape Operator and applications

There exists a holomorphic quadratic differential defined on any $H-$ surface immersed in the homogeneous space $\mathbb{E}(\kappa,\tau)$ given by U. Abresch and H. Rosenberg, called the Abresch-Rosenberg differential. However, there were no Codazzi pair on such $H-$surface associated to the Abresch-Rosenberg differential when $\tau \neq 0$. The goal of this paper is to find a geometric Codazzi pair defined on any $H-$surface in $\mathbb{E}(\kappa,\tau)$, when $\tau \neq 0$, whose $(2,0)-$part is the Abresch-Rosenberg differential. In particular, this allows us to compute a Simons' type formula for $H-$surfaces in $\mathbb{E}(\kappa,\tau)$. We apply such Simons' type formula, first, to study the behavior of complete $H-$surfaces $\Sigma$ of finite Abresch-Rosenberg total curvature immersed in $\mathbb{E}(\kappa,\tau)$. Second, we estimate the first eigenvalue of any Schr\"odinger operator $L= \Delta + V$, $V$ continuous, defined on such surfaces. Finally, together with the Omori-Yau's Maximum Principle, we classify complete $H-$surfaces in $\mathbb{E}(\kappa,\tau)$, $\tau \neq 0$, satisfying a lower bound on $H$ depending on $\kappa$ and $\tau$.