Submanifolds In Hyperbolic Spaces

Rigidity problems is one of the important problems in geometry of submanifolds,which are widely discussed by many geometers.With the deep research of the problems,the rigidity theorem can be reacted by various pinching theorems,for this kind of problem,we cain pass through the study of the first eigenvalue operator to describe the geometry of submanifolds properties.In this thesis,we use the various methods in the study of the submanifolds geometry,and combine with the analysis of various types of inequality.On the one hand,we investigate the rigidity problems of the complete noncompact hypersurface Mn with constant mean curvature in hyperbolic space Hn?1(-1),and the complete noncompact submanifolds M" with parallel mean curvature vector in hyperbolic space Hn+m+(-1).On the other hand,we study the problems of the compact oriented hypersurface Mn with constant scalar curvature in hyperbolic space Hn+1(-1).Setting |(?)|2 =|A|2-nH2 and let A is the second fundamental form of Mn,and let H is the mean curvature of Mn.The chief conclusions obtained in this thesis include the following three parts:(1)Let Mn be a complete noncompact immersed hypersurface with constant mean curvature H in hyperbolic spaces Hn+1(-1)(n ? 5),and H ? a(0<a<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 E Mn satisfy a suitable quadratic growth condition,then Mn is totally umbilical.The result in this section generalize the corresponding results in minimal circumstaice of Xia and wang[8].(2)Let AMn be a complete noncompact immersed submanifolds with parallel mean curvature vector H? in hyperbolic spaces Hn+m(-1)(n ? 6,m ? 2),and H<a(0<a<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 satisfy a suitable quadratic growth condition,then Mn is totally umbilical.The result extend and improve the corresponding ones of Antonioa[12]and Jogli[13],(3)Let Mn be a compact and oriented hypersurface with constant scalar curva-ture n(n-1)? in hyperbolic spaces Hn?1(-1),and mean curvature H ?0.Assume that the ?j1 is defined as the first eigenvalue of an Schrodinger operator,Then?j1 ?-n2[(n-1)r + n-3]min|H| + n(n-1)(r + 1)[(n-1)(r + 1)-1]1/min|H|.Moreover,the equality occurs if and only if Mn is totally umbilical but nontotally geodesic.The result extend the corresponding of Cheng[26].