Any complete non-compact Kähler surface with positive sectional curvature is biholomorphic to ℂ².
Carron:L 2 harmonic forms on non-compact manifolds,Online lecture Notes, https://www.math.sciences.univ-nantes.fr/~carron/cours.pdf
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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.