Stable And Complete Submanifolds With Constant Mean Curvature And Finite L~p-Norm Curvature In Space Forms

In this thesis, we study stable and complete submanifolds with constant mean curvature and some L^p-norm curvature in Euclidean spaces and hyperbolic spaces.Firstly, we consider a complete noncompact n-dimensional submanifold Mⁿ with parallel mean curvature vector h in an Euclidean space R^n+k. If Mⁿ has finite L^m-norm curvature for some m≥n, we prove that Mⁿ must be minimal. Moreover, if Mⁿ is strongly stable and has finite total curvature, then Mⁿ is an affine n-plane. This is a generalization of the result of Y.B. Shen and X.H. Zhu in [59] on strongly stable complete hypersurfaces with constant mean curvature in Rⁿ⁺¹:Secondly, we consider complete hypersurfaces in Mⁿ⁺¹ with constant mean curvature and prove that Mⁿ is a hyperplane if the L²-norm curvature of Mⁿ satisfies some growth condition and Mⁿ is a stable and complete hypersurface in Rⁿ⁺¹ with constant mean curvature. It is an improvement of a theorem proved by H.Alencar and M. do Carmo [2] in 1994. In addition, we obtain that Mⁿ is a hyperplane (or a round sphere) under the condition that Mⁿ is strongly stable (or weakly stable) and has finite L^p norm curvature for some p. At the same time, we prove that if Mⁿ is a complete hypersurface in Hⁿ⁺¹(-1) with constant mean curvature H > 1 and has finite L^p norm curvature for some p, then Mⁿ must be a geodesic sphere in Hⁿ⁺¹(-1).Thirdly, we consider complete hypersurfaces in Hⁿ⁺¹(-1) with constant mean curvature and finite index, we introduce the concept of k-weighted bi-Ricci curvature, which allows us to improve a result of Xu Cheng[27]. To be precise, we prove that any complete finite index hypersurface in the hyperbolic space H⁴(-1)(H⁵(-1)) with constant mean curvature H satisfying H² >64/63 (H² > 175-148 respectively) must be compact. Specially, we obtain that any complete and stable hypersurface in the hyperbolic space H⁴(-1)(H⁵(-1)) with constant mean curvature H satisfying H² >64/63 (H² >175/148 respectively) must be compact. It shows that there are no manifolds satisfying the condition of some theorems in [26] and [28].