Gap theorems for minimal submanifolds of a hyperbolic space

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 L2L2 norm of |A||A| on geodesic balls centered at some point p∈Mp∈M has less than quadratic growth and if either supx∈M⁡|A|2(x)supx∈M⁡|A|2(x) is not too large or the LnLn norm of |A||A| on M is small, here, A is the second fundamental form of M.