Abstract
In this paper we considered curvature conditions on a Kähler-Einstein surface of general type. In particular we showed that it has negative holomorphic sectional curvature if theL 2-norm of (3C 2 −C 21 )/C 21 is sufficiently small, whereC 1 andC 2 are the first and second Chern classes of the surfaces. This generalizes a result of Yau on the uniformization of Kähler-Einstein surfaces of general type and with 3C 2 −C 21 = 0. Also in the process, we obtain a necessary condition in terms of an inequality between Chern numbers for a Kähler-Einstein metric to have negative holomorphic sectional curvature.
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Cheung, CK. Chern forms, chern numbers and curvature conditions on a Kähler-Einstein surface. J Geom Anal 2, 105–119 (1992). https://doi.org/10.1007/BF02921384
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DOI: https://doi.org/10.1007/BF02921384