Schauder estimates for equations with cone metrics, I

Abstract

This is the first paper in a series to develop a linear and nonlinear theory for elliptic and parabolic equations on K\"ahler varieties with mild singularities. Donaldson has established a Schauder estimate for linear and complex Monge-Amp\`ere equations when the background K\"ahler metrics on $\mathbb{C}^n$ have cone singularities along a smooth complex hypersurface. We prove a sharp pointwise Schauder estimate for linear elliptic and parabolic equations on $\mathbb{C}^n$ with background metric $g_\beta= \sqrt{-1} ( dz_1 \wedge d\bar{z_1} + \ldots + \beta^2|z_n|^{-2(1-\beta)} dz_n \wedge d\bar{z_n}) $ for $\beta\in (0,1)$. Our results give an effective elliptic Schauder estimate of Donaldson and a direct proof for the short time existence of the conical K\"ahler-Ricci flow.