Schauder estimates for equations with cone metrics, II

Abstract

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. As an application, we prove the short-time existence of the conical Kähler-Ricci flow with conical singularities along a divisor with simple normal crossings.

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].…”

Calabi–Yau metrics with conical singularities along line arrangements

“…The space of homogeneous harmonic functions on C β 1 × C β 2 with sub-quadratic growth is explicitly identified in [17, Proposition 3.4] by using separation of variables. On the other hand, after the first version of this paper appeared, Donladson's Schauder estimates were extended to the normal crossing situation by Guo-Song [26]. Using Guo-Song's estimates we could avoid introducing a weight function at the double points of the arrangement and instead deal with them in the same way as we do with points in L × j (i.e.…”

Asymptotically conical Calabi-Yau metrics with cone singularities along a compact divisor

“…Proof. It is standard that, provided δ is small, ω ref is positive defines a metric with cone singularities along E of the required type, see [GS18]. The upper bound on the curvature follows by writing ω ref , in local coordinates where E = {z 1 .…”

“…This article is organized as follows. In Section 2 we collect together some analytic preliminaries and we recall the recent interior Schauder estimates of Guo and Song [GS18], which extend Donaldson's work [Don12] on Kähler metrics with cone singularities along a smooth divisor to the simple normal crossing situation. In Section 3 we prove Theorem 1 by using Yau's continuity path.…”