Setting and statement of the conjecture
endowed with the hyperbolic metric
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(It is not clear how Yau's name became associated with the conjecture: in unpublished correspondence with Harold Rosenberg, both Schoen and Yau identify Schoen as having postulated the conjecture). The Schoen(-Yau) conjecture has since been disproved.
The emphasis is on the existence or non-existence of an harmonic diffeomorphism, and that this property is a "one-way" property. In more detail: suppose that we consider two Riemannian manifolds M and N (with their respective metrics), and write
if there exists a diffeomorphism from M onto N (in the usual terminology, M and N are diffeomorphic). 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. In particular, is a symmetric relation:
It can be shown that the hyperbolic plane and (flat) complex plane are indeed diffeomorphic:
so the question is whether or not they are "harmonically diffeomorphic". However, as the truth of Heinz's theorem and the falsity of the Schoen–Yau conjecture demonstrate, is not a symmetric relation:
Thus, being "harmonically diffeomorphic" is a much stronger property than simply being diffeomorphic, and can be a "one-way" relation.
- Heinz, Erhard (1952). "Über die Lösungen der Minimalflächengleichung". Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt. 1952: 51–56.
- Collin, Pascal; Rosenberg, Harold (2010). "Construction of harmonic diffeomorphisms and minimal graphs". Ann. of Math. 2. 172 (3): 1879–1906. arXiv:math/0701547. doi:10.4007/annals.2010.172.1879. ISSN 0003-486X. MR2726102