Contractibility results for certain spaces of Riemannian metrics on the disc

TL;DR

This paper establishes a general criterion for the contractibility of certain subsets of Riemannian metrics on the disc, with applications to metrics with positive curvature and convex boundaries, highlighting differences from higher dimensions.

Contribution

It introduces a new contractibility criterion for Riemannian metrics on the disc, applicable to metrics with positive curvature and convex boundaries, a result not known in higher dimensions.

Findings

The space of metrics with positive Gauss curvature and convex boundary is contractible.

The same contractibility result does not extend to higher dimensions $n \,\geq\, 3$, and likely not for many $n \,\geq\, 4$.

Provides a new understanding of the topology of metric spaces on the disc.

Abstract

We provide a general contractibility criterion for subsets of Riemannian metrics on the disc. For instance, this result applies to the space of metrics that have positive Gauss curvature and make the boundary circle convex (or geodesic). The same conclusion is not known in any dimension $n\geq 3$, and (by analogy with the closed case) is actually expected to be false for many values of $n\geq 4$.