On Some Rigidity Problems About Free Boundary Hypersurfaces And Capillary Hypersurfaces

In this paper,we focus on some rigidity problems about free boundary hypersurfaces and capillary hypersurfaces.In this setting,we study the Serrin overdetermined problem in a domain with partial umbilical boundary,the higher order Alexandrovtype theorem with free boundary,the stability for two types partitioning problems and the uniqueness of capillary hypersurfaces supported on a hyperbolic horosphere.In the first part(the second and third chapters),we firstly consider a partial overdetermined boundary value problem(BVP)in a half ball B+n.We show that a domain in which this partially overdetermined problem admits a solution if and only if the domain is a spherical cap intersecting Sn-1 orthogonally.If we assume the boundary of this hypersurface is perpendicular to Sn-1,then the regularity of the overdetermined BVP can be improved further.We also generalize the results in Euclidean space into the case of general domains with partial umbilical boundary in space forms.In the second part(the fourth chapter),we firstly prove a sharp Heintze-KarcherRos inequality and Minkowski formulas for hypersurfaces with free boundary in space forms.By combining the Minkowski type formulas with the Heintze-Karcher-Ros inequality,we prove an Alexandrov-type theorem for higher order CMC hypersurfaces with free boundary in space forms.In the third part(the fifth chapter),we firstly recall a definition of Type-Ⅰ hypersurface and Wang-Xia’s results about the uniqueness of stable Type-Ⅰ hypersurfaces in a ball.Secondly,we introduce a definition of Type-Ⅱ hypersurface and show a rigidity theorem for stable Type-Ⅱ hypersurfaces in a geodesic ball in space forms as totally geodesic balls.For general ambient space and convex domain,some topological restrictions for stable Type-Ⅱ surfaces have been proved.Finally,we get a lower bound of the Morse index for Type-Ⅱ hypersurfaces in terms of their topology.In the forth part(the sixth chapter),we firstly explain the difficulties when using the argument in the fifth chapter to prove the rigidity of stable capillary hypersurfaces supported on a hyperbolic horosphere.Utilizing the properties of Killing vector field in hyperbolic space,we overcome this obstacle and prove a rigidity theorem that an immersed capillary hypersurface supported on a horosphere is stable if and only if it is umbilical.Following the same ideas above,we obtain a classification theorem for stable Type-Ⅱ hypersurfaces supported on a horosphere.