Communications in Analysis and Geometry

Volume 23 (2015)

Number 5

Four-orbifolds with positive isotropic curvature

Pages: 951 – 991

DOI: http://dx.doi.org/10.4310/CAG.2015.v23.n5.a2

Author

Hong Huang (School of Mathematical Sciences, Beijing Normal University; and Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, China)

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 discrete subgroup of the isometry group of the round cylinder $\mathbb{S}^3 \times \mathbb{R}$ on which $G$ acts freely, such that $X$ is diffeomorphic to a possibly infinite connected sum of $\mathbb{S}^4 , mathbb{RP}^4$ and 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.

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