We show that the underlying complex manifold of a complete noncompact twodimensional shrinking gradient KählerRicci soliton (M,g,X) with soliton metric g with bounded scalar curvature Rg whose soliton vector field X has an integral curve along which Rg↛0 is biholomorphic to either C×P1 or to the blowup of this manifold at one point. Assuming the existence of such a soliton on this latter manifold, we show that it is toric and unique. We also identify the corresponding soliton vector field. Given these possibilities, we then prove a strong form of the FeldmanIlmanenKnopf conjecture for finite time Type I singularities of the KählerRicci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension.
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This content will become publicly available on January 1, 2025
Schauder estimates for equations with cone metrics, II
We continue our work on 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 shorttime existence of the conical Kähler–Ricci flow with conical singularities along a divisor with simple normal crossings.
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 Award ID(s):
 2203607
 NSFPAR ID:
 10511069
 Publisher / Repository:
 Math. Sci. Publ.
 Date Published:
 Journal Name:
 Analysis & PDE
 Volume:
 17
 Issue:
 3
 ISSN:
 21575045
 Page Range / eLocation ID:
 757 to 830
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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