Convexity estimates for hypersurfaces moving by convex curvature functions

Abstract

We consider the evolution of compact hypersurfaces by fully non-linear, parabolic curvature ows for which the normal speed is given by a smooth, convex, degree one homoge-neous function of the principal curvatures. We prove that solution hypersurfaces on which the speed is initially positive become weakly convex at a singularity of the ow. The result extends the convexity estimate [HS99b] of Huisken and Sinestrari for the mean curvature ow to a large class of speeds, and leads to an analogous description of `type-II' singularities. We remark that many of the speeds considered are positive on larger cones than the positive mean half-space, so that the result in those cases also applies to non-mean-convex initial data. We consider the evolution of compact hypersurfaces by fully nonlinear, parabolic curvature flows for which the normal speed is given by a smooth, convex, degree-one homogeneous function of the principal curvatures. We prove that solution hypersurfaces on which the speed is initially positive become weakly convex at a singularity of the flow. The result extends the convexity estimate of Huisken and Sinestrari [Acta Math. 183:1 (1999), 45-70] for the mean curvature flow to a large class of speeds, and leads to an analogous description of "type-II" singularities. We remark that many of the speeds considered are positive on larger cones than the positive mean half-space, so that the result in those cases also applies to non-mean-convex initial data.

New Pinching Estimates for Inverse Curvature Flows in Space Forms

“…Some important examples of function satisfying (i)-(iv) and (v)(c) of Assumption 1.1 are f = nE 1/k k , k = 1, · · · , n, and the the power means H r = n 1− 1 r ( i κ r i ) 1/r , r > 0. More examples can be constructed as follows: If G 1 is homogeneous of degree one, increasing in each argument, and inverseconcave, and G 2 satisfies (i)-(iv) and (v)(c) of Assumption 1.1, then F = G σ 1 G 1−σ 2 satisfies (i)-(iv) and (v)(c) of Assumption 1.1 for any 0 < σ < 1 (see [5,6] for more examples on inverse concave, or convex functions).…”

Convexity estimates for surfaces moving by curvature functions

“…And finally, until recently, there was no analogue of the Huisken-Sinestrari asymptotic convexity estimate for most other flows, with the notable exception of the result of Alessandroni and Sinestrari [1], which applies to a special class of flows by functions of the mean curvature having a certain asymptotic behaviour. In a companion paper [11], the authors prove that an asymptotic convexity estimate holds for fully non-linear flows (1.1) satisfying Conditions 1.1 if, in addition, the speed f is a convex function. The main purpose of this paper is to show that an asymptotic convexity estimate holds in surprising generality for flows of surfaces; namely, the assumption that f is convex is unnecessary:…”

“…Applications of the convexity estimate are discussed in [11]. In particular, the Harnack inequality [4] yields a description of type-II singularities analogous to that of the mean curvature flow, as long as the speed f satisfies a certain concavity condition on the positive cone (this condition is satisfied, for example, if f is convex, or inverse-concave).…”

Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions

“…Results obtained in this setting have been useful in classification of singularities and the extension beyond singularities of the mean curvature flow [2,[26][27][28][29]. Recently, there has been interest in singularities of fully nonlinear curvature flows of closed nonconvex hypersurfaces [10][11][12], so it is natural to consider such flows of axially symmetric hypersurfaces as model cases. There have also been some related studies of volume-preserving mean curvature flow of axially symmetric surfaces [13][14][15][16][17][18].…”

“…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:…”