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In [[mathematics]], '''systolic inequalities for curves on surfaces''' were first studied by [[Charles Loewner]] in 1949<!-- That should be listed in the "References" section below. --> (unpublished; see remark at end of [[P. M. Pu]]'s paper in '52). Given a [[closed surface]], its [[Systolic geometry|systole]], denoted '''sys''', is defined to the least length of a loop that cannot be contracted to a point on the surface. The ''systolic area'' of a metric is defined to be the ratio area/sys<sup>2</sup>. The ''systolic ratio'' SR is the reciprocal quantity sys<sup>2</sup>/area. See also [[Introduction to systolic geometry]].
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==Torus==
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[[Image:TorusSystoleLoop.png|right|thumb|200px|Shortest loop on a torus]]
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In 1949 [[Loewner]] proved  [[Loewner's torus inequality|his inequality]] for metrics on the [[torus]] T<sup>2</sup>, namely that the systolic ratio SR(T<sup>2</sup>) is bounded above by <math>2/\sqrt{3}</math>, with equality in the flat (constant curvature) case of the equilateral torus (see [[hexagonal lattice]]).
 
==Real projective plane==
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A similar result is given by [[Pu's inequality|Pu's inequality for the real projective plane]] from 1952, due to [[Pao Ming Pu]], with an upper bound of ''π''/2 for the systolic ratio SR(RP<sup>2</sup>), also attained in the constant curvature case.
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==Klein bottle==
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[[Image:Acme klein bottle.jpg|thumb|150px|right|A hand-blown Klein Bottle (emulation)]]
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For the [[Klein bottle]] ''K'', Bavard (1986) obtained an optimal upper bound of <math> \pi/\sqrt{8}</math> for the systolic ratio:
 
:<math>\mathrm{SR}(K) \leq  \frac{\pi}{\sqrt{8}},</math>
 
based on work by Blatter from the 1960s.
 
==Genus 2==
An orientable surface of genus 2 satisfies Loewner's bound <math>\mathrm{SR}(2)\leq \tfrac{2}{\sqrt{3}}</math>, see (Katz-Sabourau '06). It is unknown whether or not every surface of positive genus satisfies Loewner's bound. It is conjectured that they all do. The answer is affirmative for genus 20 and above by (Katz-Sabourau '05).
 
==Arbitrary genus==
For a closed surface of genus ''g'', Hebda and Burago (1980) showed that the systolic ratio SR(g) is bounded above by the constant 2. Three years later, [[Mikhail Gromov (mathematician)|Mikhail Gromov]] found an upper bound for SR(g) given by a constant times
 
:<math>\frac{(\log g)^2}{g}.</math>
 
A similar ''lower'' bound (with a smaller constant) was obtained by Buser and Sarnak.  Namely, they exhibited arithmetic hyperbolic Riemann surfaces with systole behaving as a constant times <math>\log (g)</math>. Note that area is 4π(g-1) from the Gauss-Bonnet theorem, so that SR(g) behaves asymptotically as a constant times <math>\tfrac{(\log g)^2}{g}</math>.
 
The study of the asymptotic behavior for large genus <math>g</math> of the systole of hyperbolic surfaces reveals some interesting constants. Thus, [[Hurwitz surface]]s <math>\Sigma_g</math> defined by a tower of principal congruence subgroups of the [[(2,3,7) triangle group|(2,3,7) hyperbolic triangle group]] satisfy the bound
 
:<math> \mathrm{sys}(\Sigma_g) \geq \frac{4}{3} \log g,</math>
 
resulting from an analysis of the [[Hurwitz quaternion order]]. A similar bound holds for more general arithmetic [[Fuchsian group]]s.  This 2007 result by [[Mikhail Katz]], [[Mary Schaps]], and [[Uzi Vishne]] improves an inequality due to [[Peter Sarnak]] and Peter Buser in the case of arithmetic groups defined over <math>\mathbb{Q}</math>, from 1994, which contained a nonzero additive constant.  For the Hurwitz surfaces of principal congruence type, the systolic ratio SR(g) is asymptotic to
 
:<math>\frac{4}{9\pi} \frac{(\log g)^2}{g}.</math>
 
Using [[Katok's entropy inequality]], the following asymptotic ''upper bound'' for SR(g) was found in (Katz-Sabourau 2005):
 
:<math>\frac{(\log g)^2}{\pi g},</math>
 
see also (Katz 2007), p.&nbsp;85. Combining the two estimates, one obtains tight bounds for the asymptotic behavior of the systolic ratio of surfaces.
 
==Sphere==
There is also a version of the inequality for metrics on the sphere, for the invariant ''L'' defined as the least length of a closed geodesic of the metric. In '80, Gromov conjectured a lower bound of <math>1/2\sqrt{3}</math> for the ratio area/L<sup>2</sup>. A lower bound of 1/961 obtained by Croke in '88 has recently been improved by [[Alexander Nabutovsky|Nabutovsky]], Rotman, and Sabourau.
 
==See also==
*[[Differential geometry of surfaces]]
 
==References==
*{{cite journal |last=Bavard |first=C. |title={{lang|fr|Inégalité isosystolique pour la bouteille de Klein}} |journal=[[Mathematische Annalen|Math. Ann.]] |volume=274 |year=1986 |issue=3 |pages=439–441 |doi=10.1007/BF01457227 }}
*{{cite journal |last=Buser |first=P. |last2=[[Peter Sarnak|Sarnak]] |first2=P. |title=On the period matrix of a Riemann surface of large genus (With an appendix by J. H. Conway and N. J. A. Sloane) |journal=[[Inventiones Mathematicae]] |volume=117 |year=1994 |issue=1 |pages=27–56 |doi=10.1007/BF01232233 }}
*{{cite journal |last=Gromov |first=M. |title=Filling Riemannian manifolds |journal=[[Journal of Differential Geometry|J. Diff. Geom.]] |volume=18 |year=1983 |issue=1 |pages=1–147 |mr=697984 }}
*{{cite journal |last=Hebda |first=J. |title=Some lower bounds for the area of surfaces |journal=Invent. Math. |volume=65 |year=1981/82 |issue=3 |pages=485–490 |doi=10.1007/BF01396632 }}
*{{Cite book | last1=[[Mikhail Katz|Katz]] | first1=Mikhail G. | title=Systolic geometry and topology | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-4177-8 | year=2007 | volume=137 | author=}}
*{{cite journal |last=Katz |first=M. |last2=Sabourau |first2=S. |title=Entropy of systolically extremal surfaces and asymptotic bounds |journal=Ergo. Th. Dynam. Sys. |volume=25 |year=2005 |issue= 4|pages=1209–1220 |doi=10.1017/S0143385704001014 }}
*{{cite journal |last=Katz |first=M. |last2=Sabourau |first2=S. |title=Hyperelliptic surfaces are Loewner |journal=[[Proceedings of the American Mathematical Society|Proc. Amer. Math. Soc.]] |volume=134 |year=2006 |issue=4 |pages=1189–1195 |arxiv=math.DG/0407009 |doi=10.1090/S0002-9939-05-08057-3  }}
*{{cite journal |last=Katz |first=M. |last2=Schaps |first2=M. |last3=Vishne |first3=U. |title=Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups |journal=J. Differential Geom. |volume=76 |year=2007 |issue=3 |pages=399–422 |arxiv=math.DG/0505007  }}
*{{cite journal |last=Pu |first=P. M. |title=Some inequalities in certain nonorientable Riemannian manifolds |journal=[[Pacific Journal of Mathematics|Pacific J. Math.]] |volume=2 |year=1952 |pages=55–71 |mr=0048886 }}
 
{{Systolic geometry navbox}}
 
 
[[Category:Differential geometry of surfaces]]
[[Category:Systolic geometry]]

Revision as of 11:20, 5 March 2014

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