View source for Schoen–Yau conjecture

In [[Mathematics|mathematics]], the '''Schoen–Yau conjecture''' is a disproved conjecture in [[Hyperbolic geometry|hyperbolic geometry]], named after the [[Biography:Mathematician|mathematician]]s [[Biography:Richard Schoen|Richard Schoen]] and [[Biography:Shing-Tung Yau|Shing-Tung Yau]].
It was inspired by a theorem of Erhard Heinz (1952). One method of disproof is the use of [[Scherk surface]]s, as used by [[Biography:Harold Rosenberg (mathematician)|Harold Rosenberg]] and Pascal Collin (2006).

==Setting and statement of the conjecture==

Let <math>\mathbb{C}</math> be the [[Complex plane|complex plane]] considered as a [[Riemannian manifold]] with its usual (flat) Riemannian metric. Let <math>\mathbb{H}</math> denote the [[Hyperbolic space|hyperbolic plane]], i.e. the unit disc

:<math>\mathbb{H} := \{ (x, y) \in \mathbb{R}^2 | x^2 + y^2 < 1 \}</math>

endowed with the hyperbolic metric

:<math>\mathrm{d}s^2 = 4 \frac{\mathrm{d} x^2 + \mathrm{d} y^2}{(1 - (x^2 + y^2))^2}.</math>

E. Heinz proved in 1952 that there can exist no [[Harmonic map|harmonic]] [[Diffeomorphism|diffeomorphism]]

:<math>f : \mathbb{H} \to \mathbb{C}. \, </math>

In light of this theorem, Schoen conjectured that there exists no harmonic diffeomorphism

:<math>g : \mathbb{C} \to \mathbb{H}. \, </math>

(It is not clear how Yau's name became associated with the conjecture: in unpublished correspondence with Harold Rosenberg, both Schoen and Yau identify Schoen as having postulated the conjecture).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.

==Comments==

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

:<math>M \sim N\,</math>

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

:<math>M \propto N</math>

if there exists an harmonic diffeomorphism from ''M'' onto ''N''. It is not difficult to show that <math>\sim</math> (being diffeomorphic) is an [[Equivalence relation|equivalence relation]] on the objects of the category of Riemannian manifolds. In particular, <math>\sim</math> is a [[Symmetric relation|symmetric relation]]:

:<math>M \sim N \iff N \sim M.</math>

It can be shown that the hyperbolic plane and (flat) complex plane are indeed diffeomorphic:

:<math>\mathbb{H} \sim \mathbb{C},</math>

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, <math>\propto</math> is not a symmetric relation:

:<math>\mathbb{C} \propto \mathbb{H} \text{ but } \mathbb{H} \not \propto \mathbb{C}.</math>

Thus, being "harmonically diffeomorphic" is a much stronger property than simply being diffeomorphic, and can be a "one-way" relation.

==References==

* {{cite journal
|     last = Heinz
|    first = Erhard
|    title = Über die Lösungen der Minimalflächengleichung
|  journal = Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt.
|   volume = 1952
|     year = 1952
|    pages = 51–56
}}
* {{cite journal
|     last = Collin
|    first = Pascal
|author2=Rosenberg, Harold
 |    title = Construction of harmonic diffeomorphisms and minimal graphs
|  journal = Ann. of Math. |series =  2
|   volume = 172
|     year = 2010
|    issue = 3
|    pages = 1879&ndash;1906
|     issn = 0003-486X
|      doi = 10.4007/annals.2010.172.1879
|arxiv = math/0701547
}} {{MathSciNet|id=2726102}}

{{Disproved conjectures}}

{{DEFAULTSORT:Schoen-Yau conjecture}}
[[Category:Hyperbolic geometry]]

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