Ricci flow on open 4-manifolds with positive isotropic curvature

TL;DR

This paper classifies complete 4-manifolds with positive isotropic curvature using Ricci flow with surgery, extending previous results to noncompact cases and identifying their topological types.

Contribution

It extends Ricci flow classification results to noncompact 4-manifolds with positive isotropic curvature, including those with bounded geometry and no essential incompressible space form.

Findings

Manifolds are diffeomorphic to standard spaces or their connected sums.

Classification applies to noncompact, complete 4-manifolds with positive isotropic curvature.

Uses Ricci flow with surgery inspired by recent work of Bessi ext{é}res, Besson, and Maillot.

Abstract

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.