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 $(\mathrm{\Sigma },g_1)$ and $(\mathrm{\Sigma },g_2)$ when the cone angles of $g_1$ and $g_2$ are different and smaller than $\pi$. When the cone angles of $g_1$ are strictly smaller than the ones of $g_2$, this minimal diffeomorphism is unique.