Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces

Abstract In this article, we establish uniqueness and symmetry results of self-similar solutions to curvature flows by some homogeneous speed functions of principal curvatures in some warped product spaces. In particular, we proved that any compact star-shaped self-similar solution to any parabolic flow with homogeneous degree $$-1$$ - 1 (including the inverse mean curvature flow) in warped product spaces $$I \times _{\phi } M^n$$ I × ϕ M n , where $$M^n$$ M n is a compact homogeneous manifold and $$\phi '' \ge 0$$ ϕ ′ ′ ≥ 0 (including hyperbolic spaces and the anti-de Sitter-Schwarzschild space), must be a slice. The same result holds for compact self-expanders when the degree of the speed function is greater than $$-1$$ - 1 and with an additional assumption $$\phi ' \ge 0$$ ϕ ′ ≥ 0 . Furthermore, we also show that any complete non-compact mean-convex, asymptotically conical expanding self-similar solutions to the flow by a positive power of mean curvature in hyperbolic and anti-de Sitter-Schwarzschild spaces are rotationally symmetric.