A fully nonlinear flow for two-convex hypersurfaces in Riemannian manifolds

Abstract

We consider a one-parameter family of closed, embedded hypersurfaces moving with normal velocity $$G_\kappa = \big ( \sum _{i < j} \frac{1}{\lambda _i+\lambda _j-2\kappa } \big )^{-1}$$ , where $$\lambda _1 \le \cdots \le \lambda _n$$ denote the curvature eigenvalues and $$\kappa $$ is a nonnegative constant. This defines a fully nonlinear parabolic equation, provided that $$\lambda _1+\lambda _2>2\kappa $$ . In contrast to mean curvature flow, this flow preserves the condition $$\lambda _1+\lambda _2>2\kappa $$ in a general ambient manifold. Our main goal in this paper is to extend the surgery algorithm of Huisken–Sinestrari to this fully nonlinear flow. This is the first construction of this kind for a fully nonlinear flow. As a corollary, we show that a compact Riemannian manifold satisfying $$\overline{R}_{1313}+\overline{R}_{2323} \ge -2\kappa ^2$$ with non-empty boundary satisfying $$\lambda _1+\lambda _2 > 2\kappa $$ is diffeomorphic to a 1-handlebody. The main technical advance is the pointwise curvature derivative estimate. The proof of this estimate requires a new argument, as the existing techniques for mean curvature flow due to Huisken–Sinestrari, Haslhofer–Kleiner, and Brian White cannot be generalized to the fully nonlinear setting. To establish this estimate, we employ an induction-on-scales argument; this relies on a combination of several ingredients, including the almost convexity estimate, the inscribed radius estimate, as well as a regularity result for radial graphs. We expect that this technique will be useful in other situations as well.