Kähler-Einstein surface and symmetric space
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.
scientific article; zbMATH DE number 6132153