4-dimensional locally CAT(0)-manifolds with no Riemannian smoothings

TL;DR

This paper constructs 4-dimensional locally CAT(0) manifolds with nontrivial boundary at infinity knots, demonstrating they cannot admit Riemannian metrics of nonpositive curvature, thus showing a fundamental difference in geometric structures.

Contribution

The authors provide explicit examples of smooth 4-manifolds with locally CAT(0) metrics that lack Riemannian nonpositive curvature smoothings, highlighting new geometric phenomena.

Findings

Existence of 4-manifolds with nontrivial boundary at infinity knots.

Fundamental groups not realizable by Riemannian nonpositive curvature manifolds.

Counterexamples to Riemannian smoothing in 4D geometry.

Abstract

We construct examples of smooth 4-dimensional manifolds M supporting a locally CAT(0)-metric, whose universal cover X satisfy Hruska's isolated flats condition, and contain 2-dimensional flats F with the property that the boundary at infinity of F defines a nontrivial knot in the boundary at infinity of X. As a consequence, we obtain that the fundamental group of M cannot be isomorphic to the fundamental group of any Riemannian manifold of nonpositive sectional curvature. In particular, M is a locally CAT(0)-manifold which does not support any Riemannian metric of nonpositive sectional curvature.