标题： 双曲空间中完整子流形的间隙定理 显示英文标题

标题： Gap theorems for complete submanifolds in the hyperbolic space

摘要： 基于经典的西蒙斯公式，沈\cite{Shen}和林霞\cite{LX}在20世纪80年代末获得了单位球中紧致极小子流形的间隙定理。 随后由于徐\cite{Xu}、倪\cite{Ni}、云\cite{Yun}和徐-谷\cite{XuG}的影响，我们对球面或欧几里得空间中具有平行平均曲率向量场的完备子流形的间隙现象有了全面的理解。 但这类结果在双曲空间的情况下，直到最近才由Wang-Xia\cite{XiaW}、Lin-Wang\cite{LW}和 Xu-Xu\cite{XX}获得，并且到目前为止并不完全。 在本文中，首先我们继续研究双曲空间中具有平行平均曲率向量场的完备子流形的间隙定理，这些定理推广或扩展了文献中的若干结果。 其次，我们证明了一个关于双曲空间中具有常标量曲率$n(1-n)$的完备超曲面的间隙定理，该定理扩展了Bai-Luo\cite{BL2}在欧几里得空间和单位球情况下的相关结果。 双曲空间中的此类结果更为复杂，这是由于Simons公式中存在一些额外的不良项，而我们证明的主要成分之一是Lin\cite{Lin}获得的双曲空间中完备子流形的第一特征值估计。 显示英文摘要

摘要： Based on the seminal Simons' formula, Shen \cite{Shen} and Lin-Xia \cite{LX} obtained gap theorems for compact minimal submanifolds in the unit sphere in the late 1980's. Then due to the effect of Xu \cite{Xu}, Ni \cite{Ni}, Yun \cite{Yun} and Xu-Gu \cite{XuG}, we achieved a comprehensive understanding of gap phenomena of complete submanifolds with parallel mean curvature vector field in the sphere or in the Euclidean space. But such kind of results in case of the hyperbolic space were obtained by Wang-Xia \cite{XiaW}, Lin-Wang \cite{LW} and Xu-Xu \cite{XX} until relatively recently and are not quite complete so far. In this paper first we continue to study gap theorems for complete submanifolds with parallel mean curvature vector field in the hyperbolic space, which generalize or extend several results in the literature. Second we prove a gap theorem for complete hypersurfaces with constant scalar curvature $n(1-n)$ in the hyperbolic space, which extends related results due to Bai-Luo \cite{BL2} in cases of the Euclidean space and the unit sphere. Such kind of results in case of the hyperbolic space are more complicated, due to some extra bad terms in the Simons' formula, and one of main ingredients of our proofs is an estimate for the first eigenvalue of complete submanifolds in the hyperbolic space obtained by Lin \cite{Lin}.