Gap theorems for complete submanifolds in the hyperbolic space
Abstract
Based on the seminal Simons' formula, Shen Shen and Lin-Xia LX obtained gap theorems for compact minimal submanifolds in the unit sphere in the late 1980's. Then due to the effect of Xu Xu, Ni Ni, Yun Yun and Xu-Gu 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 XiaW, Lin-Wang LW and Xu-Xu 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 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 Lin.
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