On the topology of the space of pinched negatively curved metrics with finite volume and identical ends

TL;DR

This paper investigates the structure of the space of complete, finite volume, negatively curved metrics on high-dimensional manifolds, revealing it is either empty or highly disconnected when metrics share similar behavior at infinity.

Contribution

It establishes the topological complexity of the space of pinched negatively curved metrics under certain asymptotic conditions.

Findings

The space is either empty or highly non-connected.

Connectivity depends on the behavior at infinity.

Provides new insights into the topology of negatively curved metric spaces.

Abstract

We prove that the space of complete, finite volume, pinched negatively curved Riemannian metrics on a smooth high-dimensional manifold is either empty or it is highly non-connected, provided their behavior at infinity is similar.