Stability Of Capillary Hypersurfaces In Space Forms

In this thesis we study the stability of capillary hypersurfaces and the global properties of closed hypersurfaces in space forms.In the main part of the thesis we study the stability of capillary hypersurfaces in space forms. Given a Riemannian manifold( M, g), a compact hypersurface M with boundary in M is said to be capillary, if M has constant mean curvature and it intersects? M with constant contact angle. A capillary hypersurface M is said to be stable, if M is the minimum of an energy functional up to second order under any admissible volumepreserving variation. In Chapter 3 we fix M as the Euclidean unit ball. By analyzing the property of a family of important conformal transformations of the Euclidean ball acting on the capillary hypersurface, we prove the instability of capillary hypersurfaces with some symmetry in the Euclidean unit ball. In Chapter 4 we consider the case when M is a domain in Euclidean space enclosed by some hyperplanes. When the normals of these hyperplanes are linearly independent, by constructing a suitable test function, we prove that stable compact immersed capillary hypersurfaces must be a part of a standard sphere under some reasonable conditions. In Chapter 5 we consider the stability of capillary hypersurfaces in the context of manifolds with density. We generalize some well-known stability results to the manifolds with density.In the thesis we also study several global properties of closed hypersurfaces. In Chapter 6 we consider the closed embedded hypersurfaces in a certain warped product manifold M. By observing the existence of an elliptic point on the hypersurface M, we remove the convexity condition in the Alexandrov type theorem due to Brendle-Eichmair[1]concerning the higher order mean curvature of M. In Chapter 7 we consider the closed convex hypersurface M in hyperbolic space Hn+1or spherical space Sn+1. By evolving M along the unit normal to get a family of parallel hypersurfaces and analyzing the limit property of these hypersurfaces under the isoperimetric inequality constraint, we get new Alexandrov-Fenchel type inequalities.