On uniqueness of solutions to complex Monge-Amp\`ere mean field equations

TL;DR

This paper proves the uniqueness of solutions to complex Monge-Ampère mean field equations for small temperature parameters, confirming a conjecture locally and extending results globally.

Contribution

It establishes the uniqueness of solutions under specific conditions, advancing understanding of complex Monge-Ampère mean field equations and confirming a conjecture in the field.

Findings

Uniqueness of solutions for small temperature parameters

Partial confirmation of Berman and Berndtsson's conjecture

Extension of results from local to global settings

Abstract

We establish the uniqueness of solutions to complex Monge-Amp\`ere mean field equations when the temperature parameter is small. In the local setting of bounded hyperconvex domains, our result partially confirms a conjecture by Berman and Berndtsson. Our approach also extends to the global context of compact complex manifolds.