Curvature Flows For Hypersurfaces In Riemannian Manifolds And Their Geometric Applications

This thesis studies the long time existence,convergence and geometric application-s of the contracting and inverse curvature flows for hypersurfaces in Riemannian mani-folds under different convexity assumptions.In the first part of this thesis,we consider contracting flows of surfaces with positive scalar curvature in 3-dimensional hyperbolic space and strictly convex surfaces in 3-dimensional sphere.We prove that with differ-ent speed functions and powers,the evolving surfaces contract to a round point in finite time.In the second part of this thesis,we discuss the inverse curvature flow of surfaces with positive scalar curvature in 3-dimensional hyperbolic space.We prove the exis-tence and convergence of the flow when the speed is a function of Gauss curvature under some natural conditions.In the third part of this thesis,we consider the horo-convex hypersurfaces in hyperbolic space and introduce the shifted inverse curvature flow.We study the maximal existence and asymptotical behavior of the flow for horo-convex hypersurfaces and prove that the limiting shape of the solution is always round as the maximal existence time is approached.Therefore the shifted inverse curvature flow in hyperbolic space has better asymptotical behavior in contrast to the non-shifted inverse curvature flow.In the fourth part of this thesis,we use the inverse mean curvature flow to establish an optimal Minkowski type inequality,weighted Alexandrov-Fenchel inequal-ity for the mean convex star-shaped hypersurfaces in Reissner-Nordstrom-Anti-deSitter manifold and Penrose type inequality for asymptotically locally hyperbolic manifolds which can be realized as graphs over Reissner-Nordstrom-Anti-deSitter manifold.