Ricci flow with surgery in higher dimensions

• 成果类型: Article
• 署名作者: Brendle, Simon
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2018.187.1.6
• 发表日期: 2018
• 页码: 263-299
• 关键词: positive isotropic curvature; space-forms; harnack inequality; MANIFOLDS; 4-manifolds; deformation; Operators

摘要

We present a new curvature condition that is preserved by the Ricci flow in higher dimensions. For initial metrics satisfying this condition, we establish a higher dimensional version of Hamilton's neck-like curvature pinching estimate. Using this estimate, we are able to prove a version of Perelman's Canonical Neighborhood Theorem in higher dimensions. This makes it possible to extend the flow beyond singularities by a surgery procedure in the spirit of Hamilton and Perelman. As a corollary, we obtain a classification of all diffeomorphism types of such manifolds in terms of a connected sum decomposition. In particular, the underlying manifold cannot be an exotic sphere. Our result is sharp in many interesting situations. For example, the curvature tensors of CPn/2, HPn/4, Sn-k x S-k (2 <= k <= n - 2), Sn-2 x H-2, Sn-2 x R-2 all lie on the boundary of our curvature cone. Another borderline case is the pseudo-cylinder: this is a rotationally symmetric hypersurface that is weakly, but not strictly, two-convex. Finally, the curvature tensor of Sn-1 x R lies in the interior of our curvature cone.