Uniqueness and Stability in $\mathcal E(X,\omega)$

TL;DR

This paper proves the uniqueness of solutions to the Dirichlet problem for the complex Monge-Ampère equation on compact Kähler manifolds with measures vanishing on pluripolar sets, and generalizes Xing's stability theorem.

Contribution

It establishes uniqueness results for the complex Monge-Ampère equation under new measure conditions and extends a key stability theorem in the field.

Findings

Proved uniqueness for the Dirichlet problem in specified conditions.

Generalized Xing's stability theorem.

Enhanced understanding of stability and uniqueness in complex Monge-Ampère equations.

Abstract

We prove uniqueness for the Dirichlet problem for the complex Monge-Amp\`ere equation on compact K\"ahler manifolds in the case of measures vanishing on pluripolar sets. As a by-product we generalize Xing's stability theorem.