Ricci flow on open 4-manifolds with positive isotropic curvature

Details Ricci flow on open 4-manifolds with positive isotropic curvature Ricci flow on open 4-manifolds with positive isotropic curvature

2011-08-15

arxiv.org/abs/1108.2918v2

Mathematics

Differential Geometry

25 pages

Scientific paper

In this note we prove the following result: Let $X$ be a complete, connected 4-manifold with uniformly positive isotropic curvature, with bounded geometry and with no essential incompressible space form. Then $X$ is diffeomorphic to $\mathbb{S}^4$, or $\mathbb{RP}^4$, or $\mathbb{S}^3\times \mathbb{S}^1$, or $\mathbb{S}^3\widetilde{\times} \mathbb{S}^1$, or a possibly infinite connected sum of them. This extends work of Hamilton and Chen-Zhu to the noncompact case. The proof uses Ricci flow with surgery on complete 4-manifolds, and is inspired by recent work of Bessi$\grave{e}$res, Besson and Maillot.

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