Kähler-Einstein surface and symmetric space
type: Item
instance of: scholarly article
title: Kähler-Einstein surface and symmetric space
zbMATH Open document ID: 1259.53067
author: Yi Hong
author: Daguang Chen
author: Hongcang Yang
published in: Science China. Mathematics
publication date: 2013-01-29T00:00:00Z
review text: The paper is devoted to the study of the pinched holomorphic sectional curvature. Due to a work of Y. T. Siu and P. Yang, the following pinching result is known. Let \(M\) be a compact Kähler-Einstein surface with nonpositive bisectional holomorphic curvature. If there is a positive constant \(\chi < \frac{2}{3\left(1+ \sqrt{\frac{6}{11}}\right)}\) such that, for any \(p\in M\), \[ K_{av}(p)-K_{\min}(p)\leq \chi\left(K_{\max}(p)- K_{\min}(p)\right) \] on \(M\) everywhere, then \(M\) is biholomorphically isometrically isomorphic to a compact quotient manifold of the complex \(2\)-sphere (endowed with its invariant metric). Y. Hong, Z. Guan and H. Yang improved the result, proving the theorem under the weaker assumption \(\chi\leq\frac{2}{3\left(1+ \sqrt{\frac{6}{11}}\right)}\). (Note that there is a typo in the statement of this result in the paper under consideration.) In the paper under consideration, the authors improve the above result to the weaker assumption \(\chi\leq \frac{1}{2}(1-\epsilon)\). They prove furthermore that a compact Kähler-Einstein surface with nonpositive bisectional holomorphic curvature is a compact Hermitian symmetric space of dimension \(2\), assuming the curvature satisfies, for some \(t<\frac{1}{3}\), \[ K_{av}(p)-K_{\min}(p)\geq \frac{1}{2}(1+t)\left(K_{\max}(p)- K_{\min}(p)\right) \] and \[ H_{\max}(p)\leq 3\left(1+\frac{1-3t}{4t} \right) \left(3K_{av}(p) -2K_{\max}(p)-K_{\min}(p) \right), \] where \(H_{\max}(p)\) is the maximum of the bisectional curvature at \(p\). They use these results, to give pinching conditions on compact quotients of the ball of complex dimension \(2\) and the complex bidisk.
reviewed by: Walter Freyn
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32Q20
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 6132153
zbMATH Keywords: Kähler-Einstein surface
zbMATH Keywords: Hermitian symmetric space
zbMATH Keywords: holomorphic sectional curvature
zbMATH Keywords: pinching condition
MaRDI profile type: Publication
full work available at URL: s11425-011-4300-z
OpenAlex ID: W2040036098
cites work: Q3834662
cites work: Curvature characterization of compact Hermitian symmetric spaces
cites work: Q3791824
cites work: Compact Kaehler-Einstein surfaces of nonpositive bisectional curvature
cites work: Negatively \(1\over 4\)-pinched Riemannian metric on a compact Kähler manifold
cites work: Kahler-Einstein Surfaces with Nonpositive Bisectional Curvature
DOI: 10.1007/S11425-011-4300-Z
description: scientific article; zbMATH DE number 6132153