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• W1780375897 abstract "We consider a one-parameter family of closed, embedded hypersurfaces moving with normal velocity $G_kappa = big ( sum_{i 2kappa$. In contrast to mean curvature flow, this flow preserves the condition $lambda_1+lambda_2>2kappa$ 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 $overline{R}_{1313}+overline{R}_{2323} geq -2kappa^2$ with non-empty boundary satisfying $lambda_1+lambda_2 > 2kappa$ 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." @default.
• W1780375897 created "2016-06-24" @default.
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• W1780375897 date "2015-07-16" @default.
• W1780375897 modified "2023-09-27" @default.
• W1780375897 title "A fully nonlinear flow for two-convex hypersurfaces" @default.
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