Minimal diffeomorphism between hyperbolic surfaces with cone singularities

We prove the existence of a minimal diffeomorphism isotopic to the identity between two hyperbolic cone surfaces (Σ,g1) and (Σ,g2) when the cone angles of g1 and g2 are different and smaller than π. When the cone angles of g1 are strictly smaller than the ones of g2, this minimal diffeomorphism is unique.