Uniqueness of minimal diffeomorphisms between surfaces

We prove that there exists at most one minimal diffeomorphism in a given homotopy class between any two closed Riemannian surfaces. This results was previously known only under the assumption that the Riemannian metrics have constant Gaussian curvature. Along the way, we prove the New Main Inequality which substantially strengthens the classical Reich-Strebel inequality for quasiconformal maps.