Gap theorems for minimal submanifolds of a hyperbolic space

摘要

This paper provides some gap theorems for complete immersed minimal submanifolds of dimension no less than five in a hyperbolic space. Namely, we show that an n (>= 5)-dimensional complete immersed minimal submanifold M in a hyperbolic space is totally geodesic if the L-2 norm of vertical bar A vertical bar on geodesic balls centered at some point p is an element of M has less than quadratic growth and if either sup(x is an element of m) vertical bar A vertical bar(2)(x) is not too large or the L-n norm of vertical bar A vertical bar on M is small, here, A is the second fundamental form of M. (C) 2016 Elsevier Inc. All rights reserved.