Loewner's torus inequality

Loewner's torus inequality

In differential geometry, Loewner's torus inequality is an inequality due to Charles Loewner for the systole of an arbitrary Riemannian metric on the 2-torus.

tatement

In 1949 Charles Loewner proved that every metric on the 2-torus mathbb T^2 satisfies the optimal inequality

: operatorname{sys}^2 leq frac{2}{sqrt{3 ;operatorname{area}(mathbb T^2),

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 gamma_2 in dimension 2, so that Loewner's torus inequality can be rewritten as

: operatorname{sys}^2 leq gamma_2;operatorname{area}(mathbb T^2).

The inequality was first mentioned in the literature in harvtxt|Pu|1952.

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 spanned by the cube roots of unity in mathbb C.

Alternative formulation

Given a doubly periodic metric on mathbb R^2 (e.g. an imbedding in mathbb R^3 which is invariant by a mathbb Z^2 isometric action), there is a nonzero element gin mathbb Z^2 and a point pin mathbb R^2 such that operatorname{dist}(p, g.p)^2 leq frac{2}{sqrt{3 operatorname{area} (F), where F is a fundamental domain for the action, while operatorname{dist} is the Riemannian distance, namely least length of a path joining p and g . p .

Proof of Loewner's torus inequality

Loewner's torus inequality can be proved most easily by using the computational formula for the variance,

:E(X^2)-(E(X))^2=mathrm{var}(X).

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 X2 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. The variance of X can then be thought of as the isosystolic defect, analogous to the isoperimetric defect of Bonnesen's inequality. This approach therefore produces the following version of Loewner's torus inequality with isosystolic defect:

:mathrm{area}-frac{sqrt{3{2}(mathrm{sys})^2geq mathrm{var}(f),

where "f" 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

: (mathrm{sys})^2 leq gamma_2,mathrm{area}

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.

Bibliography

*Citation | last1=Katz | first1=Mikhail G. | title=Systolic geometry and topology|pages=19 | publisher=American Mathematical Society | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-4177-8 | year=2007 | volume=137

* Katz, M.; Sabourau, S.: Entropy of systolically extremal surfaces and asymptotic bounds, Ergo. Th. Dynam. Sys., 25 (2005), no. 4, 1209-1220. See arXiv|math.DG|0410312

* Katz, M.; Sabourau, S.: Hyperelliptic surfaces are Loewner, Proc. Amer. Math. Soc. 134 (2006), no. 4, 1189-1195. See arXiv|math.DG|0407009

*citation|last=Pu|first= P.M.|authorlink=P. M. Pu| title=Some inequalities in certain nonorientable Riemannian manifolds|journal= Pacific J. Math.|volume= 2 |year=1952| pages=55-71

ee also

*Pu's inequality
*Gromov's systolic inequality for essential manifolds
*Gromov's inequality for complex projective space
*Eisenstein integer (an example of a hexagonal lattice)
*systoles of surfaces


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Torus — Not to be confused with Taurus (disambiguation). This article is about the surface and mathematical concept of a torus. For other uses, see Torus (disambiguation). A torus As the distance to th …   Wikipedia

  • Charles Loewner — in 63 Born 29 May 1893(1893 05 29) Lány …   Wikipedia

  • Gromov's systolic inequality for essential manifolds — In Riemannian geometry, M. Gromov s systolic inequality for essential n manifolds M dates from 1983. It is a lower bound for the volume of an arbitrary metric on M, in terms of its homotopy 1 systole. The homotopy 1 systole is the least length of …   Wikipedia

  • Pu's inequality — [ Roman Surface representing RP2 in R3] In differential geometry, Pu s inequality is an inequality proved by P. M. Pu for the systole of an arbitrary Riemannian metric on the real projective plane RP2.tatementA student of Charles Loewner s, P.M.… …   Wikipedia

  • Inégalité torique de Loewner — En géométrie différentielle, l inégalité torique de Loewner est une inégalité établie par le mathématicien américain Charles Loewner. Elle relie la systole et la superficie d une métrique riemannienne quelconque d un tore de dimension 2.… …   Wikipédia en Français

  • Gromov's inequality for complex projective space — In Riemannian geometry, Gromov s optimal stable 2 systolic inequality is the inequality: mathrm{stsys} 2{}^n leq n!;mathrm{vol} {2n}(mathbb{CP}^n),valid for an arbitrary Riemannian metric on the complex projective space, where the optimal bound… …   Wikipedia

  • Bonnesen's inequality — is an 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 isoperimetric inequality.More precisely, consider a planar simple closed curve …   Wikipedia

  • Systolic geometry — In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner, and developed by Mikhail Gromov and others, in its arithmetic, ergodic, and topological manifestations.… …   Wikipedia

  • Introduction to systolic geometry — Systolic geometry is a branch of differential geometry, a field within mathematics, studying problems such as the relationship between the area inside a closed curve C , and the length or perimeter of C . Since the area A may be small while the… …   Wikipedia

  • Metric Structures for Riemannian and Non-Riemannian Spaces —   Author(s) Misha Gromov …   Wikipedia

Share the article and excerpts

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