Metrics with non-negative Ricci curvature on convex three-manifolds

TL;DR

This paper proves the path connectedness and contractibility of the space of smooth Riemannian metrics with non-negative Ricci curvature on the three-ball with convex boundary, and demonstrates the existence of free boundary minimal annuli in such manifolds.

Contribution

It establishes topological properties of the metric space and applies these results to prove the existence of minimal annuli in convex three-balls.

Findings

Space of metrics is path connected.

Moduli space is contractible.

Existence of free boundary minimal annuli.

Abstract

We prove that the space of smooth Riemannian metrics on the three-ball with non-negative Ricci curvature and strictly convex boundary is path connected; and, moreover, that the associated moduli space (i.e., modulo orientation-preserving diffeomorphisms of the three-ball) is contractible. As an application, using results of Maximo, Nunes, and Smith [MNS13], we show the existence of properly embedded free boundary minimal annulus on any three-ball with non-negative Ricci curvature and strictly convex boundary.