Schauder estimates for equations with cone metrics, I

2016/11/30 by Bin Guo, Jian Song, Guo, Bin +1 · 2 citations
Mathematics · #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1612.00075

openalex publication_date 2016/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ℂⁿ have cone singularities along a smooth complex hypersurface. We prove a sharp pointwise Schauder estimate for linear elliptic and parabolic equations on ℂⁿ with background metric g_β= √(-1) ( dz₁ \wedge dz₁ + … + β²|z_n|^-2(1-β) dz_n \wedge dz_n) for β∈ (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.

Citations

• On the parabolic kernel of the Schrödinger operator
• Differential equations on riemannian manifolds and their geometric applications

Cited by

• Schauder estimates on smooth and singular spaces
• Stability of the conical Kähler-Ricci flows on Fano manifolds

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