3-manifolds with(out) metrics of nonpositive curvature

TL;DR

This paper investigates which compact aspherical 3-manifolds can support Riemannian metrics of nonpositive curvature, providing conditions under which such metrics exist or do not exist, especially in the context of Haken and graph-manifolds.

Contribution

It characterizes the existence of nonpositive curvature metrics on Haken manifolds and provides explicit examples of graph-manifolds that do not admit such metrics.

Findings

Haken manifolds with boundary of zero Euler characteristic admit nonpositive curvature metrics under certain conditions.

Non-geometric Haken manifolds generally admit such metrics, but not always.

Examples of closed graph-manifolds without nonpositive curvature metrics are constructed.

Abstract

In the context of Thurstons geometrisation program we address the question which compact aspherical 3-manifolds admit Riemannian metrics of nonpositive curvature. We show that non-geometric Haken manifolds generically, but not always, admit such metrics. More precisely, we prove that a Haken manifold with, possibly empty, boundary of zero Euler characteristic admits metrics of nonpositive curvature if the boundary is non-empty or if at least one atoroidal component occurs in its canonical topological decomposition. Our arguments are based on Thurstons Hyperbolisation Theorem. We give examples of closed graph-manifolds with linear gluing graph and arbitrarily many Seifert components which do not admit metrics of nonpositive curvature.