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| {{Expert-subject|Mathematics|date=November 2008}}
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| In [[mathematics]], the '''Schoen–Yau conjecture''' is a disproved conjecture in [[hyperbolic geometry]], named after the [[mathematician]]s [[Richard Schoen]] and [[Shing-Tung Yau]].
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| It was inspired by a theorem of [[Erhard Heinz]] (1952). One method of disproof is the use of [[Scherk surface]]s, as used by Harold Rosenberg and Pascal Collin (2006).
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| ==Setting and statement of the conjecture==
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| Let <math>\mathbb{C}</math> be the [[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]]
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| :<math>\mathbb{H} := \{ (x, y) \in \mathbb{R}^2 | x^2 + y^2 < 1 \}</math>
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| endowed with the hyperbolic metric
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| :<math>\mathrm{d}s^2 = 4 \frac{\mathrm{d} x^2 + \mathrm{d} y^2}{(1 - (x^2 + y^2))^2}.</math>
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| E. Heinz proved in 1952 that there can exist no [[harmonic map|harmonic]] [[diffeomorphism]]
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| :<math>f : \mathbb{H} \to \mathbb{C}. \, </math>
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| In light of this theorem, Schoen conjectured that there exists no harmonic diffeomorphism
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| :<math>g : \mathbb{C} \to \mathbb{H}. \, </math>
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| (It is not clear how Yau's name became associated with the conjecture: in unpublished correspondence with Harold Rosenberg, both Schoen and Yau identify Schoen as having postulated the conjecture).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.
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| ==Comments==
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| 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
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| :<math>M \sim N\,</math>
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| if there exists a diffeomorphism from ''M'' onto ''N'' (in the usual terminology, ''M'' and ''N'' are diffeomorphic). Write
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| :<math>M \propto N</math>
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| 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]] on the [[object (category theory)|objects]] of the [[category (category theory)|category]] of Riemannian manifolds. In particular, <math>\sim</math> is a [[symmetric relation]]:
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| :<math>M \sim N \iff N \sim M.</math> | |
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| It can be shown that the hyperbolic plane and (flat) complex plane are indeed diffeomorphic:
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| :<math>\mathbb{H} \sim \mathbb{C},</math>
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| 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:
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| :<math>\mathbb{C} \propto \mathbb{H} \text{ but } \mathbb{H} \not \propto \mathbb{C}.</math>
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| Thus, being "harmonically diffeomorphic" is a much stronger property than simply being diffeomorphic, and can be a "one-way" relation.
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| ==References==
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| * {{cite journal
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| | last = Heinz
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| | first = Erhard
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| | title = Über die Lösungen der Minimalflächengleichung
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| | journal = Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt.
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| | volume = 1952
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| | year = 1952
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| | pages = 51–56
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| }}
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| * {{cite journal
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| | last = Collin
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| | first = Pascal
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| | coauthor = Rosenberg, Harold
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| | title = Construction of harmonic diffeomorphisms and minimal graphs
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| | journal = Ann. of Math. (2)
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| | volume = 172
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| | year = 2010
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| | issue = 3
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| | pages = 1879–1906
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| | issn = 0003-486X
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| | doi = 10.4007/annals.2010.172.1879
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| }} {{MathSciNet|id=2726102}}
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| {{DEFAULTSORT:Schoen-Yau conjecture}}
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| [[Category:Disproved conjectures]]
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| [[Category:Hyperbolic geometry]]
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| {{geometry-stub}}
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Hello. Let me introduce the author. Her name is Emilia Shroyer but it's not the most feminine name out there. For many years he's been living in North Dakota and his family members enjoys it. One of the things she loves most is to do aerobics and now she is attempting to earn money with it. Managing people is his profession.
Feel free to visit my web page :: at home std testing