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Brendle, Simon

Ricci flow with surgery in higher dimensions. (English) Zbl 1393.53055

Ann. Math. (2) 187, No. 1, 263-299 (2018).
Summary: 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 \(\mathbb{CP}^{n/2}\), \(\mathbb{HP}^{n/4}\), \(S^{n-k}\times S^k\) \((2\leq k\leq n-2)\), \(S^{n-2}\times\mathbb H^2\), \(S^{n-2}\times\mathbb R^2\) all lie on the boundary of our curvature cone. Another borderline case is the pseudo-cylinder: this is a rotationally symmetric hypersurface that 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.

Cited in 4 Documents

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions

Keywords:

Ricci flow; curvature pinching; singularity formation
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\textit{S. Brendle}, Ann. Math. (2) 187, No. 1, 263--299 (2018; Zbl 1393.53055)
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