Loomis–Whitney inequality

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(It is not clear how Yau's name became associated with the conjecture: in unpublished correspondence with Harold Rosenberg, both Schoen and Yau identify Schoen as having postulated the conjecture).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

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

$M\sim N$

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:

$M\sim N⟺N\sim M.$

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:

$\mathbb{ℂ}\propto \mathbb{ℍ}\text{ but }\mathbb{ℍ}\propto ̸\mathbb{ℂ}.$

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

References

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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang Template:MathSciNet

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