Estimates for complex singular Monge-Ampère equations via integral method

TL;DR

This work studies gradient and Laplacian estimates for singular complex Monge-Ampere equations on compact Kahler manifolds using the integral method. It develops (t, ε) approximations to handle divisorial singularities e^{-f} = |s|_D^{2κ}, and proves gradient bounds for the perturbed potentials via differential inequalities and Moser iteration, together with Sobolev inequalities relative to the perturbed metric ω_φ. It then establishes Laplacian estimates for the singular equation by formulating differential inequalities for w = e^{-Kψ} tr_ω ω_ψ and handling error terms through integration by parts, including a second Laplacian estimate under W^{1,p} regularity of e^{h}. The results cover two regimes for the singular data: Delta h bounded below and W^{1,p} regularity of e^{h}, enabling a robust a priori control essential for singular Kähler-Einstein metrics and related geometric problems.

Abstract

In this paper, we obtain gradient estimates and Laplacian estimates for the solution to the singular complex Monge-Ampère equation by applying the integral method.

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• proof : Proof of Theorem \ref{['thm gradient estimate with general F']}