Self-similar solutions of curvature flows in warped products

Abstract

In this paper we study self-similar solutions in warped products satisfying F − F =ḡ(λ(r)∂r , ν), where F is a nonnegative constant and F is in a class of general curvature functions including powers of mean curvature and Gauss curvature. We show that slices are the only closed strictly convex self-similar solutions in the hemisphere for such F . We also obtain a similar uniqueness result in hyperbolic space H 3 for Gauss curvature F and F ≥ 1.2010 Mathematics Subject Classification. 53C44, 53C40.

Self-similar solutions to fully nonlinear curvature flows by high powers of curvature

“…Without loss of generality, we assume κ 1 (x 0 ) < κ 2 (x 0 ). From the proof of Lemma 3.2 in [18], at x 0 , we have 0…”

“…Self-similar equation can be extended to more general spaces like warped product manifolds, see [18] and its references. In [18], the uniquenesses of self-similar solutions in the hemisphere were obtained when the speed function of the corresponding curvature flow satisfies Condition 1.8 in [17].…”

“…Self-similar equation can be extended to more general spaces like warped product manifolds, see [18] and its references. In [18], the uniquenesses of self-similar solutions in the hemisphere were obtained when the speed function of the corresponding curvature flow satisfies Condition 1.8 in [17]. In general, self-similar solutions to curvature flows in warped product manifolds (other than Euclidean space) do not arise from blow-up procedures, however, they provide barriers and the knowledge of them which will be important in the study of the asymptotical behavior of the corresponding curvature flows.…”

Closed self-similar solutions to flows by negative powers of curvature

“…Rigidity of hypersurfaces and uniqueness of the self-similar solutions in warped product manifolds are considered by many researcher (see [4,15,11,14,8,9] etc.). Let M n+1 = [0, r) × λ N n be a warped product manifold with metric ḡ = dr ⊗ dr + λ 2 (r)g N , where (N n , g N ) is a closed Riemannian manifold and λ(r) is a smooth and positive function.…”