A new proof of a conjecture on nonpositive Ricci curved compact Kähler-Einstein surfaces (Q1657270)

Summary: In an earlier paper, we gave a proof of the conjecture of the pinching of the bisectional curvature mentioned in those two papers of \textit{Y. Hong} et al. [Acta Math. Sin. 31, No. 5, 595--602 (1988; Zbl 0678.53060); Sci. China, Math. 54, No. 12, 2627--2634 (2011; Zbl 1259.53067)]. Moreover, we proved that any compact Kähler-Einstein surface \(M\) is a quotient of the complex two-dimensional unit ball or the complex two-dimensional plane if (1) \(M\) has a nonpositive Einstein constant, and (2) at each point, the average holomorphic sectional curvature is closer to the minimal than to the maximal. Following Siu and Yang, we used a minimal holomorphic sectional curvature direction argument, which made it easier for the experts in this direction to understand our proof. On this note, we use a maximal holomorphic sectional curvature direction argument, which is shorter and easier for the readers who are new in this direction.