Rigidity of complete minimal submanifolds in a hyperbolic space

摘要

In this paper we prove some gap theorem for complete immersed minimal submanifold of dimension no less than six or four, depending on the codimension, in a hyperbolic space Hn+m(-1). That is, we show that a high dimensional complete immersed minimal submanifold M in Hn+m(-1), is totally geodesic if the Ld norm of |A|, for some d, on geodesic balls centered at some point pM has less than quadratic growth and if either supxM|A|2 is not too large or the Ln norm of |A| on M is finite, were, A is the second fundamental form of M.