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We consider circle packings and, more generally, Delaunay circle patterns -arrangements of circles arising from a Delaunay decomposition of a finite setof points - on surfaces equipped with a complex projective structure. Motivatedby a conjecture of Kojima, Mizushima and Tan, we prove that the forgetful mapsending a complex projective structure admitting a circle packing with givennerve (resp. a Delaunay circle pattern with given nerve and intersectionangles) to the underlying complex structure is proper.

Properness for circle packings and Delaunay circle patterns on complex projective structures