Abstract:
We study deformations of hypersurfaces with normal velocity given by a smooth symmetric increasing function of the principal curvatures. Specifically we study flows where the speed is a nonlinear concave function, so that at the coordinate level the evolution is governed by a fully nonlinear parabolic PDE. For each $k \geq 3$ we construct the first flows of this kind which smoothly deform any compact $k$-convex hypersurface of Euclidean space through a family of hypersurfaces which are also $k$-convex, before forming finite-time singularities which are necessarily convex (by $k$-convexity we mean that the sum of the smallest $k$ principal curvatures is everywhere positive). That is, we show that $k$-convexity is preserved and establish an analogue of the Huisken-Sinestrari convexity estimate, which implies convexity of singularities for mean-convex mean curvature flow.
In contrast to the mean curvature flow, the fully nonlinear flows constructed here also 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 is suitably pinched.
We then employ our convexity estimate to prove Harnack and derivative estimates for the second fundamental form of solutions which are embedded. These results imply for example that sequences of rescalings about a singularity satisfy universal bounds for the second fundamental form and all of its higher derivatives on compact subsets of spacetime. The estimates are obtained by generalising an induction on scales technique introduced by Brendle-Huisken for two-convex flows to the $k$-convex setting. Our arguments apply to a general class of flows including mean-convex mean curvature flow, and in this case we recover the influential global Harnack inequality of Haslhofer-Kleiner, but without using Huisken's monotonicity formula.