Ricci flow with surgery in higher dimensions

Abstract

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 $\mathbb{C}\mathbb{P}^{n/2}$, $\mathbb{H}\mathbb{P}^{n/4}$, $S^{n-k} × S^k(2\le k\le n-2)$, $S^{n-2} × \mathbb{H}^2$, 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 $S^{n-1} × \mathbb{R}$ lies in the interior of our curvature cone.