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 C n have cone singularities along a smooth complex hypersurface. We prove a sharp pointwise Schauder estimate for linear elliptic and parabolic equations on C n with background metricfor 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.

Conical Kähler–Ricci flows on Fano manifolds

“…Since the conical Kähler-Einstein metrics play an important role in the solution of the Yau-Tian-Donaldson's conjecture which has been proved by Chen-Donaldson-Sun [6][7][8] and Tian [51], the existence and geometry of the conical Kähler-Einstein metrics have been widely concerned. The conical Kähler-Einstein metrics have been studied by Berman [1], Brendle [3], Campana-Guenancia-Pȃun [4], Donaldson [17], Guenancia-Pȃun [21], Guo-Song [22,23], Jeffres [25], Jeffres-Mazzeo-Rubinstein [26], Li-Sun [32], Mazzeo [38], Song-Wang [48], Tian-Wang [52] and Yao [58] etc. For more details, readers can refer to Rubinstein's article [44].…”

Schauder estimates for equations with cone metrics, II

“…The goal of this paper is to extend our result [20] and establish the sharp Schauder estimates for linear equations with background Kähler metric of conical singularities along divisors of simple normal crossings. We can apply and extend many techniques developed in [20], however, new estimates and techniques have to be developed because in case of conical singularities along a single smooth divisor, the difficult estimate in the conical direction can sometimes be bypassed and reduced to estimates in the regular directions, while such treatment does not work in the case of simple normal crossings. One is forced to treat regions near high codimensional singularities directly with new and more delicate estimate beyond the scope of [20].…”

“…

This is the continuation of our paper [20], to study the linear theory for equations with conical singularities. We derive interior Schauder estimates for linear elliptic and parabolic equations with a background Kähler metric of conical singularities along a divisor of simple normal crossings.

…”

“…In fact, the blow-up or perturbation techniques developed in [36,42] (also see [33,34,2,3]) are much more flexible and sharper than the classical method. The authors combined the perturbation method in [20] and geometric gradient estimates to establish sharp Schauder estimates for Laplace equations and heat equations on C n with a background flat Kähler metric of conical singularities along the smooth hyperplane {z 1 = 0} and derived explicit and optimal dependence on conical parameters.…”

Schauder estimates on smooth and singular spaces

“…This estimate plays an important role in the study of conical Kähler geometry. Its original proof due to Donaldson is by potential theory and recently, there is another proof (without potential theory) of the same estimate by Guo and Song [6]. Moreover, there is also a parabolic version of Donaldson's estimate due to Chen and Wang [4].…”