2020/07/15 by Stephen Lynch, Lynch, Stephen · 1 citation
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2007.07791
We study fully nonlinear geometric flows that deform strictly k-convex\nhypersurfaces in Euclidean space with pointwise normal speed given by a concave\nfunction of the principal curvatures. Specifically, the speeds we consider are\nobtained by performing a nonlinear interpolation between the mean and the\nk-harmonic mean of the principal curvatures. Our main result is a convexity\nestimate showing that, on compact solutions, regions of high curvature are\napproximately convex. In contrast to the mean curvature flow, the fully\nnonlinear flows considered here preserve k-convexity in a Riemannian\nbackground, and we show that the convexity estimate carries over to this\nsetting as long as the ambient curvature satisfies a natural pinching\ncondition.\n