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

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.