Uniqueness Of Self-similar Solutions To ?_k~?-curvature Flow

?_k~?-curvature flow is a family of hypersurfaces satisfying (?),where?is the principle curvature,?_k(?)is the k-th elementary symmetric polynomial of?and?is the normal vector.It includes mean curvature flow and Gauss curvature flow as special cases.In this thesis,we study the self-similar solutions to?_k~?-curvature flow in warped products.As the examples of warped products,in Euclidean space,we prove that the closed strictly convex self-similar solutions to the flow must be a round sphere;in the hemisphere,we prove that the closed strictly convex self-similar solutions to the flow must be a slice.Furthermore,for any closed strictly convex hypersurface in Euclidean space or in the hemisphere satisfying (?),where F is a class of symmetric functions including?_kand F is a non-negative constant,we prove it must be a round sphere or a slice.In the 3-dimensional hyperbolic space,we obtain a similar uniqueness result when F is Gauss curvature and F?1.