Geodesic rays and stability in the cscK problem

TL;DR

This paper establishes a connection between geodesic rays, stability conditions, and the existence of constant scalar curvature Kähler metrics, advancing the Yau-Tian-Donaldson conjecture and related stability criteria.

Contribution

It proves maximality of finite energy geodesic rays with finite Mabuchi slope and links stability conditions to the existence of cscK metrics, reducing the conjecture to a regularization problem.

Findings

Finite energy geodesic rays with finite Mabuchi slope are maximal.

Uniform K-stability and J^{K_X}-stability imply existence of cscK metrics.

A new proof of the toric uniform YTD conjecture for polarized toric manifolds.

Abstract

We prove that any finite energy geodesic ray with a finite Mabuchi slope is maximal in the sense of Berman-Boucksom-Jonsson, and reduce the proof of the uniform Yau-Tian-Donaldson conjecture for constant scalar curvature K\"{a}hler metrics to Boucksom-Jonsson's regularization conjecture about the convergence of non-Archimedean entropy functional. As further applications, we show that a uniform K-stability condition for model filtrations and the $\mathcal{J}^{K_X}$-stability are both sufficient conditions for the existence of cscK metrics. The first condition is also conjectured to be necessary. Our arguments also produce a different proof of the toric uniform version of YTD conjecture for all polarized toric manifolds. Another result proved here is that the Mabuchi slope of a geodesic ray associated to a test configuration is equal to the non-Archimedean Mabuchi invariant.