Complete 4-manifolds with uniformly positive isotropic curvature

TL;DR

This paper classifies complete 4-manifolds with uniformly positive isotropic curvature, showing they are diffeomorphic to connected sums of standard manifolds and certain orbifolds, extending previous results using Ricci flow with surgery.

Contribution

It extends classification results of 4-manifolds with positive isotropic curvature to include orbifolds and describes their topological structure via Ricci flow with surgery.

Findings

Classifies complete 4-manifolds with positive isotropic curvature.

Shows these manifolds are diffeomorphic to connected sums of standard manifolds.

Extends classification to orbifolds using Ricci flow with surgery.

Abstract

We prove the following result: Let $(X,g_0)$ be a complete, connected 4-manifold with uniformly positive isotropic curvature and with bounded geometry. Then there is a finite collection $\mathcal{F}$ of manifolds of the form $\mathbb{S}^3 \times \mathbb{R} /G$, where $G$ is a fixed point free discrete subgroup of the isometry group of the standard metric on $\mathbb{S}^3\times \mathbb{R}$, such that $X$ is diffeomorphic to a (possibly infinite) connected sum of copies of $\mathbb{S}^4,\mathbb{RP}^4$ and/or members of $\mathcal{F}$. This extends recent work of Chen-Tang-Zhu and Huang. We also extend the above result to the case of orbifolds. The proof uses Ricci flow with surgery on complete orbifolds.