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

作者：Brendle Simon^*; Huisken Gerhard

来源：Inventiones Mathematicae, 2017, 210(2): 559-613.

DOI：10.1007/s00222-017-0736-2

We consider a one-parameter family of closed, embedded hypersurfaces moving with normal velocity , where denote the curvature eigenvalues and is a nonnegative constant. This defines a fully nonlinear parabolic equation, provided that . In contrast to mean curvature flow, this flow preserves the condition 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 with non-empty boundary satisfying 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.

• 出版日期2017-11