Any complete non-compact Kähler surface with positive sectional curvature is biholomorphic to ℂ².
Stenzel:Ricci-flat metrics on the complexification of a compact rank one symmet- ric space,Manuscripta Math.,80(1993), 151–163
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Uniformisation of complete K\"ahler surfaces with positive sectional curvature
Any complete non-compact Kähler surface with positive sectional curvature is biholomorphic to ℂ².