# Schoen–Yau conjecture

In mathematics, the Schoen–Yau conjecture is a disproved conjecture in hyperbolic geometry, named after the mathematicians Richard Schoen and Shing-Tung Yau.

It was inspired by a theorem of Erhard Heinz (1952). One method of disproof is the use of Scherk surfaces, as used by Harold Rosenberg and Pascal Collin (2006).

## Setting and statement of the conjecture

Let $\mathbb {C}$ be the complex plane considered as a Riemannian manifold with its usual (flat) Riemannian metric. Let $\mathbb {H}$ denote the hyperbolic plane, i.e. the unit disc

$\mathbb {H} :=\{(x,y)\in \mathbb {R} ^{2}|x^{2}+y^{2}<1\}$ endowed with the hyperbolic metric

$\mathrm {d} s^{2}=4{\frac {\mathrm {d} x^{2}+\mathrm {d} y^{2}}{(1-(x^{2}+y^{2}))^{2}}}.$ E. Heinz proved in 1952 that there can exist no harmonic diffeomorphism

$f:\mathbb {H} \to \mathbb {C} .\,$ In light of this theorem, Schoen conjectured that there exists no harmonic diffeomorphism

$g:\mathbb {C} \to \mathbb {H} .\,$ (It is not clear how Yau's name became associated with the conjecture: in unpublished correspondence with Harold Rosenberg, both Schoen and Yau identify Schoen as having postulated the conjecture).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

$M\sim N\,$ if there exists a diffeomorphism from M onto N (in the usual terminology, M and N are diffeomorphic). Write

$M\propto N$ if there exists an harmonic diffeomorphism from M onto N. It is not difficult to show that $\sim$ (being diffeomorphic) is an equivalence relation on the objects of the category of Riemannian manifolds. In particular, $\sim$ is a symmetric relation:

$M\sim N\iff N\sim M.$ It can be shown that the hyperbolic plane and (flat) complex plane are indeed diffeomorphic:

$\mathbb {H} \sim \mathbb {C} ,$ 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, $\propto$ is not a symmetric relation:

$\mathbb {C} \propto \mathbb {H} {\text{ but }}\mathbb {H} \not \propto \mathbb {C} .$ Thus, being "harmonically diffeomorphic" is a much stronger property than simply being diffeomorphic, and can be a "one-way" relation.