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We show that every cross ratio preserving homeomorphism between boundaries ofHadamard manifolds extends to a continuous map, called circumcenter extension,provided that the manifolds satisfy certain visibility conditions. We show thatthis map is a rough isometry, whenever the manifolds admit cocompact groupactions by isometries and we improve the quasi-isometry constants provided byBiswas in the case of CAT(-1)} spaces. Finally, we provide a sufficientcondition for this map to be an isometry in the case of Hadamard surfaces.

Circumcenter extension maps for non-positively curved spaces