The Calabi flow on Kähler surfaces with bounded Sobolev constant. I
type: Item
instance of: scholarly article
title: The Calabi flow on Kähler surfaces with bounded Sobolev constant. I
zbMATH Open document ID: 1254.53097
author: X.-X. Chen
author: Weiyong He
published in: Mathematische Annalen
publication date: 2012-09-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/0710.5159
review text: This is another effort on the Calabi flow on toric surfaces. In complex geometry, a central problem is to find Kähler-Einstein metrics on Fano manifolds, i.e., compact Kähler manifolds with positive Ricci classes. For a general Kähler class on a general compact Kähler manifold, there is the similar question of finding Kähler metrics with constant scalar curvatures. We notice that any Kähler metric with constant scalar curvature in the Ricci class is automatically Kähler-Einstein. Historically, there are several methods to obtain this kind of metrics. 1. The Calabi-Aubin-Yau continuity method (see [\textit{Y.-T. Siu}, Ann. Math. (2) 127, No. 3, 585--627 (1988; Zbl 0651.53035); \textit{G. Tian}, Invent. Math. 89, 225--246 (1987; Zbl 0599.53046)]). They use \(\operatorname{Ric}(\omega _t )=t\omega +(1-t)\operatorname{Ric} (\omega _0 )\) with \(0\leq t\leq 1\). 2. The Ricci flow (see [\textit{G. Tian} and \textit{X. Zhu}, J. Am. Math. Soc. 20, No. 3, 675--699 (2007; Zbl 1185.53078)]). 3. The Calabi flow. 4. The iterating inverse Ricci operation, which the reviewer discussed with Tian on several occasions (see [\textit{Y. A. Rubinstein}, Adv. Math. 218, No. 5, 1526--1565 (2008; Zbl 1143.53065)]). More recently, Kähler-Einstein metrics (or some special metrics) on open manifolds were used to approach Kähler-Einstein metrics (see [the reviewer and the first author, Asian J. Math. 4, No. 4, 817--830 (2000; Zbl 1003.32003); \textit{C. Arezzo} and \textit{F. Pacard}, Acta Math. 196, No. 2, 179--228 (2006; Zbl 1123.53036)]; see also [the first author, \textit{C. LeBrun} and \textit{B. Weber}, J. Am. Math. Soc. 21, No. 4, 1137--1168 (2008; Zbl 1208.53072)]). For a general Kähler class, there are some special metrics generalizing the Kähler metrics with constant scalar curvature. For example, if the scalar curvature is a potential function of a holomorphic vector field, then we call the Kähler metric a Calabi extremal metric (see [\textit{E. Calabi}, Ann. Math. Stud. 102, 259--2…
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 32M25
zbMATH DE Number: 6083996
zbMATH Keywords: Calabi flow
zbMATH Keywords: toric surfaces
zbMATH Keywords: curvature bounds
zbMATH Keywords: Sobolev constant
zbMATH Keywords: scalar flat bubblings
reviewed by: Daniel Guan
MaRDI profile type: Publication
OpenAlex ID: W2017136739
arXiv ID: 0710.5159
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DOI: 10.1007/S00208-011-0723-7
description: scientific article; zbMATH DE number 6083996