Schoen-Yau conjecture

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).

etting 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} o mathbb{C}.

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

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(It is not clear how Yau's name became associated with the conjecture: in unpublished correspondence with Harold Rosenberg, both Schoen and Yau identify Schoen as having postulated the conjecture).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

Comments

It should be noted that 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} mbox{ but } mathbb{H} ot propto mathbb{C}.

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&ndash;56


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • Richard Schoen — Richard Melvin Schoen (born October 23 1950) is an American mathematician. Born in Fort Recovery, Ohio, he received his PhD from Stanford University where he is currently a Robert M. Bass Professor of Humanities and Sciences. He has two children …   Wikipedia

  • Yau, Shing-Tung — ▪ Chinese mathematician born April 4, 1949, Swatow, China       Chinese born mathematician who was awarded the Fields Medal in 1983 for his work in differential geometry.       Yau received a Ph.D. from the University of California, Berkeley, in… …   Universalium

  • Richard Schoen — Richard Melvin Schoen (gesprochen Schejn; * 23. Oktober 1950 in Celina[1] in Ohio) ist ein US amerikanischer Mathematiker, der sich mit globaler Analysis und Differentialgeometrie beschäftigt. Richard Schoen Schoen promovierte 1977 an der… …   Deutsch Wikipedia

  • Shing-Tung Yau — an der Harvard University Shing Tung Yau (kant. 丘成桐, Yale: Yau1 Sing4 Tung4; Pinyin: Qīu Chéngtóng; * 4. April 1949 in Shantou, Guangdong) ist ein …   Deutsch Wikipedia

  • Shing-Tung Yau — at Harvard Law School dining hall Born …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • Scherk surface — In mathematics, a Scherk surface is an example of a minimal surface. Sherk surfaces arise in the study of certain limiting minimal surface problems and in the study of harmonic diffeomorphisms of hyperbolic space.Construction of a simple Scherk… …   Wikipedia

  • William P. Minicozzi II — (* 13. Dezember 1967 in Bryn Mawr) ist ein US amerikanischer Mathematiker, der sich vor allem mit Minimalflächen beschäftigt. Minicozzi studierte an der Princeton University (Bachelor Abschluss 1990) und promovierte 1994 bei Richard Schoen an der …   Deutsch Wikipedia

  • Positive energy theorem — In general relativity, the positive energy theorem states that, assuming the dominant energy condition, the mass of an asymptotically flat spacetime is non negative; furthermore, the mass is zero only for Minkowski spacetime.The original proof of …   Wikipedia

  • Contributors to general relativity — General relativity Introduction Mathematical formulation Resources Fundamental concepts …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”