Local noncollapsing for complex Monge-Amp\`ere equations

TL;DR

This paper establishes a local volume noncollapsing estimate for Kähler metrics derived from complex Monge-Ampère equations under Ricci curvature bounds, aiding in diameter and gradient estimates.

Contribution

It introduces a local volume noncollapsing estimate for Kähler metrics from complex Monge-Ampère equations, assuming Ricci curvature bounds, which was not previously known.

Findings

Proves local volume noncollapsing estimate for Kähler metrics

Enables diameter and gradient estimates for solutions

Assumes local Ricci curvature lower bounds

Abstract

We prove a local volume noncollapsing estimate for K\"ahler metrics induced from a family of complex Monge-Amp\`ere equations, assuming a local Ricci curvature lower bound. This local volume estimate can be applied to establish various diameter and gradient estimate.