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

• 成果类型: Article
• 署名作者: Brendle, Simon; Huisken, Gerhard
• 署名单位: Columbia University; Eberhard Karls University of Tubingen
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-017-0736-2
• 发表日期: 2017
• 页码: 559-613
• 关键词: mean-curvature flow; inscribed radius; SINGULARITIES; 4-manifolds; interior

摘要

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.