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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The Abresch-Rosenberg shape operator and applications

Authors: José M. Espinar and Haimer A. Trejos
Journal: Trans. Amer. Math. Soc. 372 (2019), 5483-5506
MSC (2010): Primary 53A10, 53C42; Secondary 58J05
Published electronically: January 14, 2019
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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_{\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.

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José M. Espinar
Affiliation: Departamento de Matemática, Facultad de Ciencias, Universidad de Cádiz, Puerto Real 11510, Spain; and Instituto Nacional de Matemática Pura e Aplicada, 110 Estrada Dona Castorina, Rio de Janeiro 22460–320, Brazil

Haimer A. Trejos
Affiliation: Instituto de Matematica Pura y Aplicada, 110 Estrada Dona Castorina, Rio de Janeiro 22460–320, Brazil

Keywords: Constant mean curvature surfaces, homogeneous space, Codazzi pairs, finite total curvature, Simons' equation, eigenvalue estimate, pinching theorem
Received by editor(s): November 15, 2016
Received by editor(s) in revised form: July 9, 2018, and October 5, 2018
Published electronically: January 14, 2019
Additional Notes: The first author is partially supported by Spanish MEC-FEDER (Grant MTM2016-80313-P and Grant RyC-2016-19359), 2018 Leonardo Grant for Researchers and Cultural Creators, BBVA Foundation (the Foundation accepts no responsibility for the opinions, statements, and contents included in the project and/or the results thereof, which are entirely the responsibility of the authors), CNPq-Brazil (Universal Grant 402781/2016-3 and Produtividade em Pesquisa Grant 306739/2016-0), and FAPERJ-Brazil (JCNE Grant 203.171/2017-3).
The second author is supported by CNPq-Brazil.
Article copyright: © Copyright 2019 American Mathematical Society