Rigidity of submanifolds with parallel mean curvature in a hyperbolic space
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摘要: 设
$ M $ 是双曲空间中具有平行平均曲率的完备子流形,$ \Phi $ 是$ M $ 的无迹第二基本形式. 本文证明了在子流形任意测地球上$ |\Phi| $ 的$ L^2 $ 模小于二次增长条件下,$ \sup_{x\in M}|\Phi|^2(x) $ 小于某常数或者$ |\Phi| $ 的$ L^n $ 模小于某常数时,$ M $ 是全脐的, 这一结果推广了完备极小子流形的相关结果.Abstract: Let$ M $ be a complete submanifold with parallel mean curvature in a hyperbolic space and$ \Phi $ be the traceless second fundamental form of$ M $ . In this paper, it is shown that the submanifold is totally umbilical if the$ L^2 $ norm of$ |\Phi| $ has less than quadratic growth on any geodesic ball of$ M $ and either$ \sup_{x\in M}|\Phi|^2(x) $ is less than some constant or$ L^n $ norm of$ |\Phi| $ is less than some constant. This is a generalization of the results on complete minimal submanifolds. -
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