Gradient estimates for Donaldson's equation on a compact K\"ahler manifold

TL;DR

This paper establishes a gradient estimate for Donaldson's equation on compact Kähler manifolds, utilizing uniform bounds and the ABP maximum principle, advancing understanding of complex geometric PDEs.

Contribution

It provides the first direct gradient estimate for Donaldson's equation on compact Kähler manifolds using maximum principle techniques.

Findings

Gradient estimate derived from uniform bounds

Applicable to both elliptic and parabolic forms of the equation

Enhances analytical tools for complex geometric PDEs

Abstract

We prove a gradient estimate for Donaldson's equation \[\omega\wedge(\chi+\sqrt{-1}\partial\overline{\partial}\varphi)^{n-1}=e^F(\chi+\sqrt{-1}\partial\overline{\partial}\varphi)^n\] (and its parabolic analog) on an $n$-dimensional compact K\"ahler manifold $(M,\omega)$ with another Hermitian metric $\chi$ directly from the uniform upper bounds for $tr_\omega\chi_\varphi$ and Alexandrov-Bakelman-Pucci (ABP) maximum principle.