Characterizations of umbilic hypersurfaces in warped product manifolds

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

“…There have been lots of results on the uniqueness of self-similar solutions for the positive power case β > 0, especially for mean curvature flow and Gauss curvature flow (see, for example, [10,6,5,7]). For the negative power case β < 0, Ma and the author [9] show some uniqueness results when F = H k is the k-th mean curvature and ambient space M n+1 can be some warped product manifolds which includes R n+1 . In this paper, we extend the result of negative power case to some general curvature function F with suitable conditions.…”

“…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.…”

“…Now, we briefly recall the methods in [9] and compare them to the proof of Theorem 2. For example, let us consider the case M n+1 = R n+1 and M n satisfies (2)…”

“…where H = 1 n (κ 1 + ... + κ n ) is normalized mean curvature. One method in [9] is based on an integral inequality…”