In
differential geometry, Loewner's torus inequality is an
inequality
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
due to
Charles Loewner
Charles Loewner (29 May 1893 – 8 January 1968) was an American mathematician. His name was Karel Löwner in Czech and Karl Löwner in German.
Karl Loewner was born into a Jewish family in Lany, about 30 km from Prague, where his father Sig ...
. It relates the
systole and the
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...
of an arbitrary
Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
on the
2-torus.
Statement
In 1949
Charles Loewner
Charles Loewner (29 May 1893 – 8 January 1968) was an American mathematician. His name was Karel Löwner in Czech and Karl Löwner in German.
Karl Loewner was born into a Jewish family in Lany, about 30 km from Prague, where his father Sig ...
proved that every metric on the 2-
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
satisfies the optimal inequality
:
where "sys" is its
systole, i.e. least length of a noncontractible loop. The constant appearing on the right hand side is the
Hermite constant in dimension 2, so that Loewner's torus inequality can be rewritten as
:
The inequality was first mentioned in the literature in .
Case of equality
The boundary case of equality is attained if and only if the metric is flat and homothetic to the so-called ''equilateral torus'', i.e. torus whose group of deck transformations is precisely the
hexagonal lattice
The hexagonal lattice or triangular lattice is one of the five two-dimensional Bravais lattice types. The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an angle of 120° ...
spanned by the cube roots of unity in
.
Alternative formulation
Given a doubly periodic metric on
(e.g. an imbedding in
which is invariant by a
isometric action), there is a nonzero element
and a point
such that
, where
is a fundamental domain for the action, while
is the Riemannian distance, namely least length of a path joining
and
.
Proof of Loewner's torus inequality
Loewner's torus inequality can be proved most easily by using the computational formula for the variance,
:
Namely, the formula is applied to the
probability measure defined by the measure of the unit area flat torus in the conformal class of the given torus. For the random variable ''X'', one takes the conformal factor of the given metric with respect to the flat one. Then the expected value E(''X''
2) of ''X''
2 expresses the total area of the given metric. Meanwhile, the expected value E(''X'') of ''X'' can be related to the systole by using
Fubini's theorem
In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...
. The variance of ''X'' can then be thought of as the isosystolic defect, analogous to the isoperimetric defect of
Bonnesen's inequality
Bonnesen's inequality is an inequality (mathematics), inequality relating the length, the area, the radius of the incircle and the radius of the circumcircle of a Jordan curve. It is a strengthening of the classical isoperimetry, isoperimetric ine ...
. This approach therefore produces the following version of Loewner's torus inequality with isosystolic defect:
:
where ''ƒ'' is the conformal factor of the metric with respect to a unit area flat metric in its conformal class.
Higher genus
Whether or not the inequality
:
is satisfied by all surfaces of nonpositive
Euler characteristic is unknown. For
orientable surfaces of genus 2 and genus 20 and above, the answer is affirmative, see work by Katz and Sabourau below.
See also
*
Pu's inequality for the real projective plane
*
Gromov's systolic inequality for essential manifolds In the mathematical field of Riemannian geometry, M. Gromov's systolic inequality bounds the length of the shortest non-contractible loop on a Riemannian manifold in terms of the volume of the manifold. Gromov's systolic inequality was proved in 1 ...
*
Gromov's inequality for complex projective space
In Riemannian geometry, Gromov's optimal stable 2- systolic inequality is the inequality
: \mathrm_2^n \leq n!
\;\mathrm_(\mathbb^n),
valid for an arbitrary Riemannian metric on the complex projective space, where the optimal bound is attained
b ...
*
Eisenstein integer (an example of a hexagonal lattice)
*
Systoles of surfaces
References
*
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