2015/07/16 by Brendle, S., Huisken, G. · 3 citations
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1507.04651
We consider a one-parameter family of closed, embedded hypersurfaces moving with normal velocity Gκ= ( ∑i < j (1)/(λi+λj-2κ) )-1, where λ1 ≤ \hdots ≤ λn denote the curvature eigenvalues and κ is a nonnegative constant. This defines a fully nonlinear parabolic equation, provided that λ1+λ2>2κ. In contrast to mean curvature flow, this flow preserves the condition λ1+λ2>2κ 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 R1313+R2323 ≥ -2κ2 with non-empty boundary satisfying λ1+λ2 > 2κ 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.