Open manifolds with uniformly positive isotropic curvature

Abstract

We prove the following result: Let $(M,g_0)$ be a complete noncompact manifold of dimension $n\geq 12$ with isotropic curvature bounded below by a positive constant, with scalar curvature bounded above, and with injectivity radius bounded below. Then there is a finite collection $\mathcal{X}$ of spherical $n$-manifolds and manifolds of the form $\mathbb{S}^{n-1} \times \mathbb{R} /G$, where $G$ is a discrete subgroup of the isometry group of the round cylinder $\mathbb{S}^{n-1}\times \mathbb{R}$, such that $M$ is diffeomorphic to a (possible infinite) connected sum of members of $\mathcal{X}$. This extends a recent work of Huang. The proof uses Ricci flow with surgery on open orbifolds with isolated singularities.Comment: arXiv admin note: text overlap with arXiv:1909.1226