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On a generalized Monge-Ampère equation on closed almost Kähler surfaces

Ken Wang, Zuyi Zhang, Tao Zheng, Peng Zhu

TL;DR

This work extends Calabi-Yau theory to closed almost Kähler surfaces by formulating a generalized Monge-Ampère equation $(\omega + \mathcal{D}_J^+(\varphi))^2 = e^{f}\omega^2$ and solving for a smooth potential $\varphi$. The central tool is the operator $\mathcal{D}_J^+$, a generalization of $\partial_J\bar{\partial}_J$, enabling a Yau-type existence and uniqueness theory in the almost Kähler setting. The authors develop a full a priori estimate program, including a $C^0$ bound and higher-order estimates, via the continuity method and local Darboux coordinates, to obtain a smooth solution with $\varphi \in C^{\infty}(M,J)_0$. As an application, they verify Donaldson's tameness conjecture for certain taming symplectic forms on almost complex 4-manifolds by producing uniform bounds along the solution path. The results broaden Calabi-Yau theory beyond Kähler geometry and provide new tools for the study of tamed almost complex 4-manifolds.

Abstract

We show the existence and uniqueness of solutions to a generalized Monge-Ampère equation on closed almost Kähler surfaces, where the equation depends only on the underlying almost Kähler structure. As an application, we prove Donaldson's conjecture for tamed almost complex 4-manifolds.

On a generalized Monge-Ampère equation on closed almost Kähler surfaces

TL;DR

This work extends Calabi-Yau theory to closed almost Kähler surfaces by formulating a generalized Monge-Ampère equation and solving for a smooth potential . The central tool is the operator , a generalization of , enabling a Yau-type existence and uniqueness theory in the almost Kähler setting. The authors develop a full a priori estimate program, including a bound and higher-order estimates, via the continuity method and local Darboux coordinates, to obtain a smooth solution with . As an application, they verify Donaldson's tameness conjecture for certain taming symplectic forms on almost complex 4-manifolds by producing uniform bounds along the solution path. The results broaden Calabi-Yau theory beyond Kähler geometry and provide new tools for the study of tamed almost complex 4-manifolds.

Abstract

We show the existence and uniqueness of solutions to a generalized Monge-Ampère equation on closed almost Kähler surfaces, where the equation depends only on the underlying almost Kähler structure. As an application, we prove Donaldson's conjecture for tamed almost complex 4-manifolds.

Paper Structure

This paper contains 4 sections, 18 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Uniqueness Theorem for the generalized Monge Ampère equation
  4. Existence Theorem for the generalized Monge Ampère equation

Key Result

Theorem 1.1

Suppose that $(M,\omega,J,g)$ is a closed almost Kähler surface, then there exists a unique solution, $\varphi \in C^{\infty}(M,J)_0$, of the generalized Monge-Ampère equation for $\varphi$ satisfying $\omega + \mathcal{D}_J^+(\varphi) > 0$, where $f$ is any smooth real function with and there is a $C^{\infty}$$a\ priori$ bound of $\varphi$ depending only on $\omega,J,$ and $f$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Conjecture 1.2
  • Corollary 1.3
  • Remark 1.4
  • Proposition 2.1
  • Definition 2.2
  • Proposition 2.3: DLZ10
  • Proposition 2.4: Lejmi10E
  • Definition 2.5
  • Proposition 2.6
  • ...and 24 more