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 was no Codazzi pair on such an H-surface associated with 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. We denote such a pair as (I,II_{\rm AR}), were I is the usual first fundamental form of the surface and II_{\rm AR} is the Abresch-Rosenberg second fundamental form. In particular, this allows us to compute a Simons' type equation for H-surfaces in {\mathbb{E}(\kappa , \tau )}. We apply such Simons' type equation, 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 Schrödinger operator L= \Delta + V, V continuous, defined on such surfaces. Finally, together with the Omori-Yau maximum principle, we classify complete H-surfaces in {\mathbb{E}(\kappa , \tau )}, \tau \neq 0, satisfying a lower bound on H depending on \kappa , \tau , and an upper bound on the norm of the traceless II_{\rm AR}, a gap theorem.