Volume Preserving Curvature Flow Research Articles

EVOLUTION OF CONVEX HYPERSURFACES BY A FULLY NONLINEAR MIXED VOLUME PRESERVING CURVATURE FLOW

In this paper we study the evolution of closed convex hypersurfacesunder the mixed volume preserving curvature flow in Euclidean spacewith the speed given by reversed function that is symmetric and ho-mogeneous of degree one. We prove that the hypersurfaces preserveconvexity under the flow, the maximum existence time is infinite andthe hypersurfaces asymptotically approach to sphere

EVOLUTION OF CONVEX HYPERSURFACES BY A FULLY NONLINEAR MIXED VOLUME PRESERVING CURVATURE FLOW

In this paper we study the evolution of closed convex hypersurfacesunder the mixed volume preserving curvature flow in Euclidean spacewith the speed given by reversed function that is symmetric and ho-mogeneous of degree one. We prove that the hypersurfaces preserveconvexity under the flow, the maximum existence time is infinite andthe hypersurfaces asymptotically approach to sphere

Volume preserving flow by powers of symmetric polynomials in the principal curvatures

We study a volume preserving curvature flow of convex hypersurfaces, driven by a power of the k-th elementary symmetric polynomial in the principal curvatures. Unlike most of the previous works on related problems, we do not require assumptions on the curvature pinching of the initial datum. We prove that the solution exists for all times and that the speed remains bounded and converges to a constant in an integral norm. In the case of the volume preserving scalar curvature flow, we can prove that the hypersurfaces converge smoothly to a round sphere.

More Mixed Volume Preserving Curvature Flows

We extend the results of McCoy (Calc Var Partial Differ Equ 24:131–154, 2005) to include several new cases where convex surfaces evolve to spheres under mixed volume preserving curvature flows, using recent results for unconstrained curvature flows and new regularity arguments in the constrained flow setting. We include results for speeds that are degree 1 homogeneous in the principal curvatures and indicate how, with sufficient curvature pinching conditions on the initial hypersurfaces, some results may be extended to speed homogeneous of degree \(\alpha >1\). In particular, these extensions require lower speed bounds that are obtained here without using estimates for equations of porous medium type, in contrast to most previous work.

Convex hypersurfaces evolving by volume preserving curvature flows

We consider the evolution of a closed convex hypersurface in euclidean space under a volume preserving flow whose speed is given by a positive power of the mean curvature. We prove that the solution exists for all times and converges to a sphere. The result does not assume the curvature pinching properties or the restrictions on the dimension that were usually required in the previous literature. The proof of the convergence exploits the monotonicity of the isoperimetric ratio satisfied by this class of flows.

Volume preserving curvature flows in Lorentzian manifolds

Let N be a (n+1)-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface and a curvature function F, which is either the mean curvature, the root of the second symmetric polynomial or a curvature function of class (K*). We consider curvature flows with curvature function F and a volume preserving term and prove long time existence of the flow and exponential convergence of the graphs in the C-infinity topology to a hypersurface of constant F-curvature, provided there are barriers. Furthermore we examine stability properties and foliations of constant F-curvature hypersurfaces.

Volume-preserving flow by powers of the mth mean curvature

We consider the evolution of a closed convex hypersurface under a volume preserving curvature flow. The speed is given by a power of the mth mean curvature plus a volume preserving term, including the case of powers of the mean curvature or of the Gauss curvature. We prove that if the initial hypersurface satisfies a suitable pinching condition, the solution exists for all times and converges to a round sphere.

Mixed volume preserving curvature flows

Mixed volume preserving curvature flows