Width and flow of hypersurfaces by curvature functions

Abstract

Abstract. In this paper, we generalize a well-known estimate of ColdingMinicozzi for the extinction time of convex hypersurfaces in Euclidean space evolving by their mean curvature to a much broader class of parabolic curvature flows.It is well known that a convex closed hypersurface in R n+1 evolving by its mean curvature remains convex and vanishes in finite time. In [1], B. Andrews showed that the same holds for a much broader class of evolutions. T. H. Colding and W. P. Minicozzi in [7] give a bound on extinction time for mean curvature flow in terms of an invariant of the initial hypersurface that they call the width. In this paper, we generalize this estimate to the class evolutions considered by Andrews: All functions F in this class are assumed to be either concave or convex. In the concave case, we require an a priori pinching condition on the initial hypersurface M 0 . In the case of convexity a pinching condition is not needed, and we can take C 0 = 1 above. This class is of interest since it contains many classical flows as particular examples, such as the n-th root of the Gauss curvature, mean curvature, and hyperbolic mean curvature flows. The key to arriving at our estimate on extinction time is a uniform estimate on the rate of change of the width, for which preservation of convexity of the evolving hypersurface is fundamental.We also study flows which are similar to those considered by Andrews in [1], except that we allow for higher degrees of homogeneity. The motivation for this consideration is that the degree 1 homogeneity condition on the flows given in Andrews' paper excludes some naturally arising evolutions, such as powers of mean curvature, for which preservation convexity and extinction time estimates are known

Curvature contraction of convex hypersurfaces by nonsmooth speeds

“…Such an estimate was developed in [9] for the mean curvature flow. A similar estimate was shown in [8] for the class of fully nonlinear flows of convex hypersurfaces [2]. We remark that the class in [2] has been broadened [3][4][5][6] and a width estimate also holds in these cases, since uniform parabolicity of these flows implies F may be estimated from below by H , important in the proof of the width estimate.…”

“…For a proof of (5.1) in the case of smooth F , we refer the reader to [25]. We note that the estimate (5.1) also holds in the case where F satisfies Conditions 1.1 but is not smooth; a proof in this setting may be found in [8].…”

Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions

“…Similar flows of hypersurfaces have been considered before, particularly flows by Gauss curvature and powers of the mean curvature and often for surfaces, usually in the context of convex initial data or translating solutions [1,4,5,[7][8][9]19,23,30,31,[38][39][40][41].…”

“…These conditions, sometimes with some adjustments, have been used before in curvature contraction flows of convex hypersurfaces [3,[6][7][8][9]19,23] and recently in flows of closed hypersurfaces not necessarily convex [10][11][12]35]. For our main result concerning the behaviour of solutions at the maximal existence time, we require the following additional structure condition on F:…”