Schoen–Yau conjecture

Key takeaways

Schoen–Yau conjecture belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Schoen–Yau conjecture to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Schoen–Yau conjecture from memory before moving on to harder problems.

Reference excerpt

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

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

endowed with the hyperbolic metric

d s 2 = 4 d x 2 + d y 2 ( 1 − ( x 2 + y 2 ) ) 2 . {\displaystyle \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 : H → C . {\displaystyle f:\mathbb {H} \to \mathbb {C} .\,}

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

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(It is not clear how Yau's name became associated with the conjecture: in unpublished correspondence with Harold Rosenberg, both Schoen and Yau identify Schoen as having postulated the conjecture).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

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

M ∝ N {\displaystyle M\propto N}

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

M ∼ N ⟺ N ∼ M . {\displaystyle M\sim N\iff N\sim M.}

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

H ∼ C , {\displaystyle \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, ∝ {\displaystyle \propto } is not a symmetric relation:

C ∝ H but H ∝̸ C . {\displaystyle \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.

References Heinz, Erhard (1952). "Über die Lösungen der Minimalflächengleichung". Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt. 1952: 51–56. Collin, Pascal; Rosenberg, Harold (2010). "Construction of harmonic diffeomorphisms and minimal graphs". Ann. of Math. 2. 172 (3): 1879–1906. arXiv:math/0701547. doi:10.4007/annals.2010.172.1879. ISSN 0003-486X. MR 2726102

Worked examples

Example 1 — a first encounter with Schoen–Yau conjecture

Start with the simplest possible case. Write down what Schoen–Yau conjecture claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Schoen–Yau conjecture before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Schoen–Yau conjecture ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Schoen–Yau conjecture

In research: Schoen–Yau conjecture appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Schoen–Yau conjecture in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Schoen–Yau conjecture is common in secondary-school and first-year university syllabi. It links to neighbouring topics Disproved conjectures, Hyperbolic geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Schoen–Yau conjecture outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Schoen–Yau conjecture in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Schoen–Yau conjecture means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Schoen–Yau conjecture out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Schoen–Yau conjecture in simple terms?

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).

Why does Schoen–Yau conjecture matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Schoen–Yau conjecture?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Schoen–Yau conjecture.