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 (3)

• Bessel Functions, Heat Kernel and the Conical Kähler-Ricci Flow
• The Ricci flow on the sphere with marked points
• A remark on Kahler metrics with conical singularities along a simple normal crossing divisor

Cited by in corpus (8)

• Degenerating Kähler-Einstein cones, locally symmetric cusps, and the Tian-Yau metric
• Construction of constant scalar curvature Kähler cone metrics
• Schauder estimates on smooth and singular spaces
• Schauder estimates on products of cones
• ALE Calabi-Yau metrics with conical singularities along a compact divisor
• Stability of the conical Kähler-Ricci flows on Fano manifolds
• Metric contraction of the cone divisor by the conical Kähler-Ricci flow
• Calabi-Yau metrics with cone singularities along intersecting complex lines: the unstable case