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 $g_{\beta }=\sqrt{-1}(d{z}_1\wedge d\overline{z_1}+\dots +{\beta }^2|z_n{|}^{-2(1-\beta )}d{z}_n\wedge d\overline{z_n})$ 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 Kähler-Ricci flow.

References in corpus (1)

• The Ricci flow on the sphere with marked points

Cited by in corpus (1)

• Stability of the conical Kähler-Ricci flows on Fano manifolds