Complete Submanifolds With Constant Mean Curvature In Hyperbolic Spaces

In this paper,we mainly study the rigidity problems of the complete submanifold Mn with constant mean curvature in hyperbolic space Hn+m(-1).Setting |?|2:=|A|2-(?)and let A is the second fundamental form of M.We get the following theorems:(1)Let Mn be an n(n>5)-dimensional complete noncompact immersed submanifold with constant mean curvature H in hyperbolic spaces Hn+m(-1).Assume that the Ln norm of |?| on Mn less than a positive constant,and the L2 norm of |?| on geodesic balls centered at some point p ?Mn has less than a suitable quadratic growth,then Mn is totally umbilical.(2)Let Mn be an n(n>5)-dimensional complete noncompact immersed submanifold with constant mean curvature H in hyperbolic spaces Hn+m(-1).if the Ln norm of |?|on Mn less than a positive constant,and the L2 norm of |?| on geodesic balls centered at some point p ? Mn has less than a suitable quadratic growth,then Mn is a hypersurface in hyperbolic spaces Hn+1(-1).(3)Let Mn be a n(n ? 4)-dimensional complete noncompact,hypersurface has two distinct constant principal curvatures,and constant mean curvature H is constant.if the the Ln norm of |?| on Mn is finite and sup x?M|?|2(x)is not too large,then Mn is a isoparametric hypersurfaces.Further abtain Mn is homogeneous.