Uniqueness Of Self-similar Solutions To σ_k~α-curvature Flow

σ_k~α-curvature flow is a family of hypersurfaces satisfying (?),whereκis the principle curvature,σ_k(κ)is the k-th elementary symmetric polynomial ofκandνis the normal vector.It includes mean curvature flow and Gauss curvature flow as special cases.In this thesis,we study the self-similar solutions toσ_k~α-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 includingσ_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.