More Mixed Volume Preserving Curvature Flows

Abstract

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 α>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.ABSTRACT. We extend the results of [28] 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 speeds homogeneous of degree α > 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.

New Pinching Estimates for Inverse Curvature Flows in Space Forms

“…Their proof relies on the Gauss map parametrization of the flow (3.1). A similar computation also appeared in [22] for the mixed volume preserving flow in Euclidean space. Our proof of the pinching estimate along the inverse curvature flow is inspired by their argument but instead of using the Gauss map parametrization of the flow we will prove this estimate directly using the evolution equations (2.8) and (2.9).…”

Non-homothetic convex ancient solutions for flows by high powers of curvature

“…In this setting, in fact, there are derivative estimates, see e.g. Section 4 in [24] and the references therein, yielding regularity of the flow as long as the curvature is bounded. It follows that the solution of (2.1) exists up to a finite maximal time at which either the norm |h| 2 blows up, or else the curvatures reach the boundary of the cone Γ.…”

Quermassintegral preserving curvature flow in Hyperbolic space

“…In previous work, the convergence of solutions as t → ∞ was deduced using either the monotonicity of curvature pinching ratios [20,24,25,26,27,13,14,17,28] or of isoperimetric ratios [3,29,11,12,8]. In our situation for general F and α, neither of these arguments is available.…”

“…In 2001, the first named author [3] studied volume preserving anisotropic mean curvature flows in R n+1 and obtained a similar result. Later, McCoy [24,25,26,27] studied some mixed volume preserving curvature flow driven by homogeneous of degree one curvature functions. For higher homogeneity, by imposing a strong pinching assumption on the initial hypersurface, Cabezas-Rivas and Sinestrari [14] proved convergence results for the flow (1.1) in R n+1 with Ψ = E α/k k where k = 1, · · · , n and α > 1.…”