Any complete non-compact Kähler surface with positive sectional curvature is biholomorphic to ℂ².
Yau:Open problems in geometry,Lectures on Differential Geometry,1 (1994), 365–404
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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.