Real-analytic geodesics in the Mabuchi space of K\"ahler metrics and quantization

TL;DR

This paper proves that quantized Bergman geodesics converge to Mabuchi geodesics for real-analytic initial data in short time, addressing a conjecture and discussing the non-existence of solutions to boundary value problems.

Contribution

It demonstrates convergence of quantized geodesics to Mabuchi geodesics for real-analytic data and explores the non-existence of boundary value problem solutions.

Findings

Convergence of quantized Bergman geodesics to Mabuchi geodesics in short time.

Partial resolution of Rubinstein and the last author's conjecture.

Non-existence of solutions to the boundary value problem in general.

Abstract

We prove the convergence of quantized Bergman geodesics to the Mabuchi geodesics for the initial value problem, in the case of real-analytic initial data and in short time. This partially solves a conjecture of Y. Rubinstein and the last author. We also argue against the existence of a solution to the boundary value problem, generically in real-analytic regularity.