Abstract
We study fully nonlinear geometric flows that deform strictly k-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained by performing a nonlinear interpolation between the mean and the k-harmonic mean of the principal curvatures. Our main result is a convexity estimate showing that, on compact solutions, regions of high curvature are approximately convex. In contrast to the mean curvature flow, the fully nonlinear flows considered here preserve k-convexity in a Riemannian background, and we show that the convexity estimate carries over to this setting as long as the ambient curvature satisfies a natural pinching condition.
Citation
Stephen Lynch. "Convexity estimates for hypersurfaces moving by concave curvature functions." Duke Math. J. 171 (10) 2047 - 2088, 15 July 2022. https://doi.org/10.1215/00127094-2022-0011
Information