A Compact Kähler Surface of Negative Curvature Not Covered by the Ball

Abstract

The uniformization theorem for Riemann surfaces says that a simply connected Riemann surface must be the Riemann sphere, the whole complex plane, or the open unit disc. In the higher dimensional case there is no such simple trichotomy, because generic slight perturbations of the ball give rise to complex manifolds no two of which are biholomorphic [1]. However, it is conjectured that with reasonable curvature assumptions a similar trichotomy exists in the higher dimensional case. Corresponding to the case of the Riemann sphere, one has the Frankel conjecture that a compact Kihler manifold of positive sectional curvature must be biholomorphic to the complex projective space. This conjecture was proved in the dimension 2 case by Andreotti-Frankel [2] and in the dimension 3 case by Mabuchi [5]. Very recently the general case was proved independently by Mori [6] using algebraic geometry of positive characteristic and by Siu-Yau [11] using the complex-analyticity of harmonic maps. (Mori's result is stronger than the result of Siu-Yau. Mori's result assumes only that the manifold has ample tangent bundle, whereas the result of Siu-Yau assumes that the manifold has positive holomorphic bisectional curvature.) Corresponding to the case of the complex plane, one has the conjecture that a noncompact complete Kahler manifold of positive sectional curvature must be biholomorphic to some C. Or, more generally, a noncompact simply connected complete Kahler manifold with sectional curvature K > -A/r'+ or even with the weaker assumption K ? -lk(r) with k(r) > 0 and