Kähler-Einstein surface and symmetric space (Q1934542)

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.