The Abresch–Rosenberg shape operator and applications

Abstract

There exists a holomorphic quadratic differential defined on any H H -surface immersed in the homogeneous space E ( κ , τ ) {\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 H -surface associated with the Abresch–Rosenberg differential when τ ≠ 0 \tau eq 0 . The goal of this paper is to find a geometric Codazzi pair defined on any H H -surface in E ( κ , τ ) {\mathbb {E}(\kappa , \tau )} , when τ ≠ 0 \tau eq 0 , whose ( 2 , 0 ) (2,0) -part is the Abresch–Rosenberg differential. We denote such a pair as ( I , I I AR ) (I,II_\textrm {AR}) , were I I is the usual first fundamental form of the surface and I I AR II_\textrm {AR} is the Abresch–Rosenberg second fundamental form. In particular, this allows us to compute a Simons’ type equation for H H -surfaces in E ( κ , τ ) {\mathbb {E}(\kappa , \tau )} . We apply such Simons’ type equation, first, to study the behavior of complete H H -surfaces Σ \Sigma of finite Abresch–Rosenberg total curvature immersed in E ( κ , τ ) {\mathbb {E}(\kappa , \tau )} . Second, we estimate the first eigenvalue of any Schrödinger operator L = Δ + V L= \Delta + V , V V continuous, defined on such surfaces. Finally, together with the Omori–Yau maximum principle, we classify complete H H -surfaces in E ( κ , τ ) {\mathbb {E}(\kappa , \tau )} , τ ≠ 0 \tau eq 0 , satisfying a lower bound on H H depending on κ \kappa , τ \tau , and an upper bound on the norm of the traceless I I AR II_\textrm {AR} , a gap theorem.