Ricci Flow on Open 4-Manifolds with Positive Isotropic Curvature

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 S 4 , or RP 4 , or S 3 × S 1 , or S 3 ×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 Bessières, Besson and Maillot.

Four-orbifolds with positive isotropic curvature

“…This is a continuation of our previous work [Hu1] on classifying open 4-manifolds with uniformly positive isotropic curvature. Following Chen-Tang-Zhu's work [CTZ] we'll remove the condition of no essential incompressible space form in [Hu1] and obtain the following Theorem 1.1.…”

“…In [Hu1] we have established a crucial weak openness (w.r.t. time) property of the canonical neighborhood condition for the noncompact manifold case (see Claim 1 in the proof of Proposition 3.6 there), which can be easily extended to the noncompact orbifold case.…”

“…Following Chen-Tang-Zhu's work [CTZ] we'll remove the condition of no essential incompressible space form in [Hu1] and obtain the following Theorem 1.1. Let (X, g 0 ) be a complete, connected 4-manifold with uniformly positive isotropic curvature and with bounded geometry.…”

“…90-92 of [K]) according to the edges of G, one obtains a smooth 4-manifold diffeomorphic to X. Note that we do not assume that the elements in X are closed manifolds or the graph is locally finite; compare [BBM], [Hu1]. By the way, note that by Cerf's theorem any diffeomorphism of the 3-sphere extends to the 4-ball.…”

“…But as already noticed by [H97] (see also [CTZ]), to do surgery on a 4-manifold with positive isotropic curvature (which contains essential incompressible space form) will lead in general to orbifolds (with isolated singularities). So we need to extend the construction in [Hu1] to the orbifold situation. Our treatment of surgery procedure is somewhat different from that in [CTZ] which considers the case of compact 4-orbifolds with at most isolated singularities.…”