On rigidity phenomena of compact surfaces in homogeneous 3-manifolds

Hu, Zejun; Lyu, Dongliang; Wang, Jing

doi:10.1090/S0002-9939-2015-12356-8

On rigidity phenomena of compact surfaces in homogeneous $3$-manifolds
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by Zejun Hu, Dongliang Lyu and Jing Wang PDF

Proc. Amer. Math. Soc. 143 (2015), 3097-3109 Request permission

Abstract:

Let $E(\kappa ,\tau )$ be the $3$-dimensional homogeneous Riemannian manifold with isometry group of dimension $4$, where $\kappa$ is the curvature of the basis and $\tau$ the bundle curvature, which satisfy $\kappa -4\tau ^2\not =0$. A special case of $E(\kappa ,\tau )$ is the Berger sphere that is also denoted by $\mathbb {S}^3_b(\kappa ,\tau )$. In this paper, surfaces of $E(\kappa ,\tau )$ are studied. As the main result, rigidity theorems in terms of the second fundamental form are established for compact (minimal) surfaces of $\mathbb {S}^3_b(\kappa ,\tau )$.

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