One method of disproof is the use of Scherk surfaces, as used by Harold Rosenberg and Pascal Collin (2006).
denote the hyperbolic plane, i.e. the unit disc endowed with the hyperbolic metric E. Heinz proved in 1952 that there can exist no harmonic diffeomorphism In light of this theorem, Schoen conjectured that there exists no harmonic diffeomorphism (It is not clear how Yau's name became associated with the conjecture: in unpublished correspondence with Harold Rosenberg, both Schoen and Yau identify Schoen as having postulated the conjecture).
Write if there exists an harmonic diffeomorphism from M onto N. It is not difficult to show that
(being diffeomorphic) is an equivalence relation on the objects of the category of Riemannian manifolds.
However, as the truth of Heinz's theorem and the falsity of the Schoen–Yau conjecture demonstrate,