Research On Several Problems Of Submanifolds In Hyperbolic Space

In this paper, by computing the Laplace of the square of the length of the second fundamental form and introducing a self-adjoint operator and using Stokes Theorem and Hopf Theorem, we obtained some pinching theorems and rigidity theorems for hypersurfaces and submanifolds in hyperbolic space. It mainly concludes the following:1. We obtained the pinching theorem for hypersurfaces with constant mean curvature in hyperbolic space about the square of the length of the second fundamental form. The rigidity theorem for hypersurfaces with constant mean curvature and nonnegative sectional curvature was also proved.2. We obtained the pinching theorem for hypersurfaces with constant scalar curvature in hyperbolic space. We also proved the rigidity theorem for hypersurfaces with constant scalar curvature and nonnegative sectional curvature in hyperbolic space.3. We obtained the pinching theorems for submanifolds with parallel mean curvature in hyperbolic space. In this paper, we spread the similar results in sphere to hyperbolic space.4. We obtained two kinds of sufficient condition for submanifolds with parallel mean curvature becoming totally umbilical hypersurfaces that lie in a totally geodesic submanifold in hyperbolic space.5. We obtained a kind of sufficient condition for 2-harmonic submanifolds becoming minimal submanifolds in hyperbolic space and an integral inequality like Simons integral inequality.