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

TL;DR

This paper proves a near-optimal version of Donaldson's conjecture linking geodesic stability to the existence of constant scalar curvature Kähler metrics, by analyzing the geometry of Mabuchi geodesic rays and convexity properties.

Contribution

It advances understanding of Kähler geometry by establishing a key stability conjecture through detailed metric and convexity analysis of Mabuchi space.

Findings

Proves a near-optimal form of Donaldson's geodesic stability conjecture.

Analyzes the metric geometry of Mabuchi geodesic rays.

Shows uniform convexity properties of the space of Kähler metrics.

Abstract

We establish the essentially optimal form of Donaldson's geodesic stability conjecture regarding existence of constant scalar curvature K\"ahler 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 K\"ahler metrics.