Abstract
In an earlier paper, we gave a proof of the conjecture of the pinching of the bisectional curvature mentioned in those two papers of Hong et al. of 1988 and 2011. 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.
Highlights
In [1], the authors conjectured that any compact Kähler–Einstein surface with negative 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, Kav − Kmin ≤ a(Kmax − Kmin ), M is a quotient of the complex ball
We conjectured in [3] that M is a quotient of the complex ball if a = 12
Summary
In [1], the authors conjectured that any compact Kähler–Einstein surface with negative 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, Kav − Kmin ≤ a(Kmax − Kmin ), M is a quotient of the complex ball. We prove the existence of the ball-like points, as we did in the second section in [6], by using a different but similar function. We again use Hong-Cang Yang’s function and a different but similar calculation with respect to the maximal direction instead of the minimal direction. We give some detailed calculations in the Appendix as the last section of this paper
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