Geodesic stability, the space of rays, and uniform convexity in Mabuchi geometry

Abstract

We establish the essentially optimal form of Donaldson's geodesic stability conjecture regarding existence of constant scalar curvature Kähler metrics. We carry this out by exploring in detail the metric geometry of Mabuchi geodesic rays, and the uniform convexity properties of the space of Kähler metrics.

arXiv commentv2: Changes in presentation. References added. v3: We show that approximation is possible with C^11 rays with converging radial K-energy. As an application, we prove the C^11 uniform geodesic stability conjecture. v4: we shifted the main focus of the paper towards geodesic stability, reflected in the presentation and a new title