Ricci flow with surgery in higher dimensions

TL;DR

This paper develops a new curvature condition preserved by Ricci flow in higher dimensions, enabling the extension of the flow with surgery and leading to a classification of certain manifolds.

Contribution

It introduces a novel curvature condition and extends Hamilton and Perelman's Ricci flow techniques to higher dimensions with surgery.

Findings

Established a higher-dimensional version of Hamilton's neck-like curvature estimate.

Proved a higher-dimensional Perelman canonical neighborhood theorem.

Classified manifolds with the new curvature condition, excluding exotic spheres.

Abstract

We present a new curvature condition which 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 $\mathbb{CP}^{n/2}$, $\mathbb{HP}^{n/4}$, $S^{n-k} \times S^k$ ($2 \leq k \leq n-2$), $S^{n-2} \times \mathbb{H}^2$, $S^{n-2} \times \mathbb{R}^2$ all lie on the boundary of our curvature cone. Another borderline case is the pseudo-cylinder: this is a rotationally symmetric hypersurface which is weakly, but not strictly, two-convex. Finally, the curvature tensor of $S^{n-1} \times \mathbb{R}$ lies in the interior of our curvature cone.