Flow by Powers of the Gauss Curvature
نویسندگان
چکیده
We prove that convex hypersurfaces in Rn+1 contracting under the flow by any power α > 1 n+2 of the Gauss curvature converge (after rescaling to fixed volume) to a limit which is a smooth, uniformly convex self-similar contracting solution of the flow. Under additional central symmetry of the initial body we prove that the limit is the round sphere.
منابع مشابه
Mean Curvature Flow with Convex Gauss Image
We study the mean curvature flow of complete space-like submanifolds in pseudo-Euclidean space with bounded Gauss image, as well as that of complete submanifolds in Euclidean space with convex Gauss image. By using the confinable property of the Gauss image under the mean curvature flow we prove the long time existence results in both cases. We also study the asymptotic behavior of these soluti...
متن کاملSe p 20 02 Gauss Maps of the Mean Curvature Flow
Let F : Σ n × [0, T) → R n+m be a family of compact immersed sub-manifolds moving by their mean curvature vector. We show the Gauss maps γ : (Σ n , g t) → G(n, m) form a harmonic heat flow with respect to the time-dependent induced metric g t. This provides a more systematic approach to investigate higher codimension mean curvature flows. A direct consequence is any convex function on G(n, m) p...
متن کاملGauss Maps of the Mean Curvature Flow
Let F : Σ n × [0, T) → R n+m be a family of compact immersed submanifolds moving by their mean curvature vectors. We show the Gauss maps γ : (Σ n , g t) → G(n, m) form a harmonic heat flow with respect to the time-dependent induced metric g t. This provides a more systematic approach to investigating higher codimension mean curvature flows. A direct consequence is any convex function on G(n, m)...
متن کاملSpacelike hypersurfaces with constant $S$ or $K$ in de Sitter space or anti-de Sitter space
Let $M^n$ be an $n(ngeq 3)$-dimensional complete connected and oriented spacelike hypersurface in a de Sitter space or an anti-de Sitter space, $S$ and $K$ be the squared norm of the second fundamental form and Gauss-Kronecker curvature of $M^n$. If $S$ or $K$ is constant, nonzero and $M^n$ has two distinct principal curvatures one of which is simple, we obtain some charact...
متن کامل“bubbling” of the Prescribed Curvature Flow on the Torus
Abstract. By a classical result of Kazdan-Warner, for any smooth signchanging function f with negative mean on the torus (M,gb) there exists a conformal metric g = egb Gauss curvature Kg = f , which can be obtained from a minimizer u of Dirichlet’s integral in a suitably chosen class of functions. As shown by Galimberti, these minimizers exhibit “bubbling” in a certain limit regime. Here we sha...
متن کامل