A fully nonlinear flow for two-convex hypersurfaces

Abstract

We consider a one-parameter family of closed, embedded hypersurfaces moving with normal velocity $G_{\kappa }=({\sum }_{i<j}\frac{1}{{\lambda }_i+{\lambda }_j-2\kappa }{)}^{-1}$, where $λ_1 \leq \hdots \leq λ_n$ 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.

to appear in Invent Math

References in corpus (3)

• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Mean curvature flow with surgery
• An inscribed radius estimate for mean curvature flow in Riemannian manifolds

Cited by in corpus (3)

• Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes
• Ancient solutions for Andrews' hypersurface flow
• Neck Detection for Two-Convex Hypersurfaces Embedded in Euclidean Space undergoing Brendle-Huisken G-Flow