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On uniqueness of solutions to complex Monge-Ampère mean field equations

Chinh H. Lu, Trong-Thuc Phung

Abstract

We establish the uniqueness of solutions to complex Monge-Ampère 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.

On uniqueness of solutions to complex Monge-Ampère mean field equations

Abstract

We establish the uniqueness of solutions to complex Monge-Ampère 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.

Paper Structure

This paper contains 4 sections, 14 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Monge-Ampère mean field equations in hyperconvex domains
  3. Stability and uniqueness on compact Kähler manifolds
  4. Monge-Ampère equations on compact Hermitian manifolds

Key Result

Theorem 1.1

Assume $(X,\omega)$ is a compact Kähler manifold of dimension $n$, and $f$ is a probability density function on $X$ which belongs to $L^p(X,\omega^n)$, for some $p>1$. Then there exists $\gamma_0>0$, depending on $(X,\omega,n,p, \|f\|_p)$, such that for all $\gamma \in (0,\gamma_0)$, the equation eq

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: The class $\mathcal{E}_p$, $\mathcal{F}$, and $\mathcal{T}_0$
  • Lemma 2.2: Maximal solution
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4: Mixed Monge-Ampère inequalities
  • proof
  • Theorem 2.5
  • ...and 19 more