Gap theorems for complete submanifolds in the hyperbolic space

TL;DR

This paper extends gap theorems for complete submanifolds with parallel mean curvature in hyperbolic space and proves a new gap theorem for hypersurfaces with constant scalar curvature, building on and generalizing prior results.

Contribution

It generalizes existing gap theorems to hyperbolic space and introduces a new gap theorem for hypersurfaces with constant scalar curvature in hyperbolic space.

Findings

Extended gap theorems for submanifolds with parallel mean curvature in hyperbolic space.

Proved a gap theorem for hypersurfaces with constant scalar curvature in hyperbolic space.

Utilized eigenvalue estimates to handle complex terms in the Simons' formula.

Abstract

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}.