Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature

We consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity. We prove that the ratio of circumradius to inradius is bounded by a function of the circumradius with limit 1 at zero. We apply this result to the motion of hypersurfaces by arbitrary speeds which are smooth homogeneous functions of the principal curvatures of degree greater than one. For smooth, strictly convex initial hypersurfaces with ratio of principal curvatures sufficiently close to one at each point, we prove that solutions remain smooth and strictly convex and become spherical in shape while contracting to points in finite time. Recently several papers have considered the flow of convex hypersurfaces by speeds which are homogeneous functions of the principal curvatures of degree α > 1. Under suitable pinching conditions on the curvature of the initial hypersurface, the aim is to prove that solutions become spherical as they contract to points. This behaviour has been established for a wide range of flows where the speed is homogeneous of degree 1 in the principal curvatures, including the mean curvature flow (H1), the flow by nth root of Gauss curvature (Ch1), square root of scalar curvature (Ch2), and a large family of other speeds (A1,A3,A4). The first such result with degree of homogeneity higher than 1 was due to Ben Chow (Ch1), and concerned flow by powers of the Gauss curvature. He proved that flow by K β with β ≥ 1/n produces a spherical limiting shape provided the initial hypersurface is sufficiently pinched, i n the sense that hij ≥ C(β)Hgij. Later such results were proved by Schulze (S3) for powers of the mean curvature, by Alessandroni and Sinestrari for powers of the scalar curvature (A,AS), and by Cabezas-Rivas and Sinestrari (CRS) for normalized flows by powers of elementary symmetric functions. A feature of the results mentioned above is that the flows all have some divergence structure, a point which is used crucially in deriving sufficient regularity of solutions to deduce the existence of a smooth limiting hypersurface: In (Ch1) the divergence structure was used to deduce pinching of the principal curvatures using integral estimates in a manner similar to (H1). In (S3), (A), and (AS) the curvature pinching was proved using maximum principle arguments, but the divergence structure was still needed because higher regularity of solutions was established using results for divergence form porous-medium equations. To date we are not aware of any work which provides Holder continuity for solutions of porous medium equations in non-divergence form without assuming regularity of the coefficients. In this paper we provide a geometric estimate which circumvents this difficulty: We prove in Section 3 that if the ratios of principal curvatures at each point are bounded in terms of the maximum principal curvature by a function which approaches one at infinity, then the ratio of circumradius to inradius is as close to one as desired if the circumradius is sufficiently small. Using