Conformal Killing Vector Fields In Space Forms And The Rigidity Of Hypersurfaces

In this paper,we will consider three types of conformal Killing vector fields and use them to prove a couple of geometric inequalities in space forms,including the various weighted Perez type inequality on sub-static free boundary hypersurfaces and the weighted Heintze-Karcher-Ros type inequality on the capillary hypersurface.Furthermore,for all the inequalities above,we will discuss the rigidity of the hypersurface on which the equalities hold.For the rigidity of capillary hypersurface in hyperbolic space,we give a new proof of the stability of a compact,free boundary hypersurfaces with constant mean curvature supported on a totally geodesic hyperbolic plane at first,and prove that the geodesic spherical caps are the only stable ones.Then we use the Heintze-Karcher-Ros type inequality in hyperbolic space to prove an Alexandrov-type theorem that any compact embedded capillary hypersurfaces with constant mean curvature can only be umbilical with non-zero principal curvature with the contacting angle θ between 0 and π/2.Also,we apply a Minkowski type formula to prove that any compact immersed capillary hypersurfaces with a constant quotient of k-th and l-th mean curvature can only be totally umbilical hypersurfaces with non-zero principal curvature.For the closed sub-static Riemannian manifold,we prove the weighted Lellis-Topping type inequality with respect to the Lovelock curvatures.Also,we prove the stability of the closed hypersurface with constant k-th shifted mean curvature in hyperbolic spaces.