In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Schoen–Yau conjecture is a disproved conjecture in
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For a ...
, named after the
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
s
Richard Schoen
Richard Melvin Schoen (born October 23, 1950) is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984 and his works on harmonic maps.
Earl ...
and
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and professor emeritus at Harvard University. Until 2022, Yau was the William Caspar ...
.
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
Harold Rosenberg (February 2, 1906 – July 11, 1978) was an American writer, educator, philosopher and art critic. He coined the term Action Painting in 1952 for what was later to be known as abstract expressionism. Rosenberg is best known for h ...
and Pascal Collin (2006).
Setting and statement of the conjecture
Let
be the
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
considered as a
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
with its usual (flat) Riemannian metric. Let
denote the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P' ...
, i.e. the
unit disc
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1:
:D_1(P) = \.\,
The closed unit disk around ''P'' is the set of points whose d ...
:
endowed with the hyperbolic metric
:
E. Heinz proved in 1952 that there can exist no
harmonic
In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the ''fundamental frequency'' of a periodic signal. The fundamental frequency is also called the ''1st har ...
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.
Definit ...
:
In light of this theorem, Schoen conjectured that there exists no harmonic diffeomorphism
:
(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
:
if there exists a diffeomorphism from ''M'' onto ''N'' (in the usual terminology, ''M'' and ''N'' are diffeomorphic). Write
:
if there exists an harmonic diffeomorphism from ''M'' onto ''N''. It is not difficult to show that
(being diffeomorphic) is an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...
on the
objects of the
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
of Riemannian manifolds. In particular,
is a
symmetric relation
A symmetric relation is a type of binary relation. Formally, a binary relation ''R'' over a set ''X'' is symmetric if:
: \forall a, b \in X(a R b \Leftrightarrow b R a) ,
where the notation ''aRb'' means that .
An example is the relation "is equ ...
:
:
It can be shown that the hyperbolic plane and (flat) complex plane are indeed diffeomorphic:
:
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,
is not a symmetric relation:
:
Thus, being "harmonically diffeomorphic" is a much stronger property than simply being diffeomorphic, and can be a "one-way" relation.
References
*
*
{{DEFAULTSORT:Schoen-Yau conjecture
Disproved conjectures
Hyperbolic geometry