Chern forms, chern numbers and curvature conditions on a Kähler-Einstein surface

Abstract

In this paper we considered curvature conditions on a Kahler-Einstein surface of general type. In particular we showed that it has negative holomorphic sectional curvature if theL2-norm of (3C2 −C12)/C12 is sufficiently small, whereC1 andC2 are the first and second Chern classes of the surfaces. This generalizes a result of Yau on the uniformization of Kahler-Einstein surfaces of general type and with 3C2 −C12 = 0. Also in the process, we obtain a necessary condition in terms of an inequality between Chern numbers for a Kahler-Einstein metric to have negative holomorphic sectional curvature.