On bisectional nonpositively curved compact Kähler–Einstein surfaces

Abstract

On Bisectional Nonpositively Curved Compact K¨ahler Einstein Surfaces Daniel Guan October 20, 2016 In this note we explain that the conjecture of the pinching of the bisec- tional curvature mentioned in [HGY] and [CHY] is proved by a combination of the arguments from the proofs of the Theorem 1.2 in [CHY], the The- orem 2 in [HGY] and the Proposition 4 in [SY]. Moreover, we prove that any compact K¨ahler-Einstein surface M is a quotient of the complex two dimensional unit ball or the complex two dimensional plane if (1) M has nonpositive Einstein constant and (2) at each point, the average holomorphic sectional curvature is closer to the minimal than to the maximal. Introduction In [SY] the authors conjectured that any compact K¨ahler surface with nega- tive bisectional curvature is a quotient of the complex two dimensional unit ball. They proved that there is a number a ∈ (1/3, 2/3) such that if at every point P , K av − K min ≤ a[K max − K min ], then M is a quotient of the complex ball. Here, K min (K max , K av ) is the minimal (maximal, aver- age) of the holomorphic sectional curvature. The number a they obtained is a 1/3. Therefore, we conjectured that M is a quotient of the complex ball if a = 2 1 . In general, we believe that we Key Words and Phrases: K¨ ahler-Einstein metrics, compact complex surfaces, bisec- tional curvature curvature, pinching of the curvatures. Math. Subject Classifications: 53C21, 53C55, 32M15, 32Q20. For this part, it is due to Professor Hong. Notice that he was the first author there.