The Abresch–Rosenberg shape operator and applications

Abstract

There exists a holomorphic quadratic differential defined on any H− surface immersed in the homogeneous space E(κ, τ ) given by U. Abresch and H. Rosenberg [1,2], called the Abresch-Rosenberg differential. However, there were no Codazzi pair on such H−surface associated to the Abresch-Rosenberg differential when τ = 0. The goal of this paper is to find a geometric Codazzi pair defined on any H−surface in E(κ, τ ), when τ = 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 E(κ, τ ). We apply such Simons' type formula, first, to study the behavior of complete H−surfaces Σ of finite Abresch-Rosenberg total curvature immersed in E(κ, τ ). Second, we estimate the first eigenvalue of any Schrödinger operator L = ∆ + V , V continuous, defined on such surfaces. Finally, together with the Omori-Yau's Maximum Principle, we classify complete H−surfaces in E(κ, τ ), τ = 0, satisfying a lower bound on H depending on κ and τ . 2010 MSC: 53C42, 58J50.

Simons type formulas for surfaces in Sol_3 and applications

“…By using this result, M. Batista [3] introduced an operator S on a CMC surface of S 2 × R or H 2 × R that satisfies the classical Codazzi property, and then obtained a Simons type equation using S instead of the shape operator A. This result was then generalized by J. M. Espinar and H. A. Trejos [9] to all other spaces where Abresch-Rosenberg differentials do exist.…”