This article demonstrates the existence for all time of the nonparametric spacelike mean curvature flow with contact angle boundary condition, where the boundary manifold is a convex cylinder. We also consider the asymptotic behavior of the flow and prove that the flow converges to a spacelike hypersurface (unique up to translation) moving at a constant speed if solutions to an elliptic system exist.Mathematics Subject Classification. Primary 53C40, 53C44; Secondary 35K55.

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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.

We consider closed orientable hypersurfaces in a wide class of warped product manifolds which include space forms, deSitter-Schwarzschild and Reissner-Nordström manifolds. By using a new integral formula or Brendle's Heintze-Karcher type inequality, we present some new characterizations of umbilic hypersurfaces. These results can be viewed as generalizations of the classical Jellet-Liebmann theorem and the Alexandrov theorem in Euclidean space.

In this paper, we investigate closed strictly convex hypersurfaces in ℝ n + 1 {\mathbb{R}^{n+1}} which shrink self-similarly under a large family of fully nonlinear curvature flows by high powers of curvature. When the speed function is given by powers of a homogeneous of degree 1 and inverse concave function of the principal curvatures with power greater than 1, we prove that the only such hypersurfaces are round spheres. We also prove that slices are the only closed strictly convex self-similar solutions to such curvature flows in the hemisphere 𝕊 + n + 1 {\mathbb{S}^{n+1}_{+}} with power greater than or equal to 1.