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 $(\Sigma, g_1)$ and $(\Sigma, g_2)$ 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.

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Received 3 March 2015

Accepted 8 May 2017

Published 12 November 2019