On $\mathcal{D}^+_J$ operator on higher dimensional almost Kähler manifolds

TL;DR

This work extends Calabi-Yau–type Monge-Ampère theory to higher-dimensional almost Kähler manifolds by introducing the generalized operator $\mathcal{D}^+_J$ and the auxiliary $\mathcal{W}_J$ framework, enabling a robust elliptic setup for the generalized Monge-Ampère equation $(\omega+\mathcal{D}^+_J(f))^n=e^F\omega^n$. It establishes local and global solvability with $C^{\infty}$ a priori estimates depending only on $\omega$, $J$, and $F$, and proves a Calabi-type existence result via the continuity method. The paper then derives and discusses Calabi conjecture-type results for almost Kähler manifolds, including existence and uniqueness of Hermite-Einstein almost Kähler metrics in negative and zero first Chern class cases, while outlining open questions for the positive case. Together, these results generalize key aspects of Kähler geometry to the nonintegrable setting, providing a foundation for further study of extremal metrics, Hermite-Einstein structures, and geometric flows in almost Kähler geometry.

Abstract

In this paper, we define $\mathcal{D}^+_J$ operator that is a generalized $\partial\bar{\partial}$ operator on higher dimensional almost Kähler manifolds. In terms of $\mathcal{D}^+_J$ operator, we study $\bar{\partial}$-problem in almost Kähler geometry and the generalized Monge-Ampère equation on almost Kähler manifolds. Similarly to the Kähler case, we obtain $C^\infty$ a priori estimates for the solution of the generalized Monge-Ampère equation on the almost Kähler manifold $(M,ω,J)$ depended only on $ω$ and $J$. Then as done in Kähler geometry, we study Calabi conjecture for almost Kähler manifolds. Finally, we will pose some questions in almost Kähler geometry.

Theorems & Definitions (45)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Theorem 2.5
• Corollary 2.6
• Proposition 2.7