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 Kähler varieties with mild singularities. Donaldson has established a Schauder estimate for linear and complex Monge-Ampère equations when the background Kähler 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_β= \sqrt{-1} ( dz_1 \wedge d\bar{z_1} + \ldots + β^2|z_n|^{-2(1-β)} dz_n \wedge d\bar{z_n}) $ for $β\in (0,1)$. Our results give an effective elliptic Schauder estimate of Donaldson and a direct proof for the short time existence of the conical Kähler-Ricci flow.