The Abresch–Rosenberg shape operator and applications

Abstract:

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 $(I,II_\textrm {AR})$, were $I$ is the usual first fundamental form of the surface and $II_\textrm {AR}$ 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 $II_\textrm {AR}$, a gap theorem.