  We prove that every 1-Lipschitz map from a closed metric surface onto a
closed Riemannian surface that has the same area is an isometry. If we replace
the target space with a non-smooth surface, then the statement is not true and
we study the regularity properties of such a map under different geometric
assumptions. Our proof relies on a coarea inequality for continuous Sobolev
functions on metric surfaces that we establish, and which generalizes a recent
result of Esmayli--Ikonen--Rajala.
