Loewner's torus inequality

Charles Loewner in 1963

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

Contents

StatementEdit

 
Shortest loop on a torus

In 1949 Charles Loewner proved that every metric on the 2-torus   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 Pu (1952).

Case of equalityEdit

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  .

Alternative formulationEdit

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 inequalityEdit

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. 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:

 

where ƒ is the conformal factor of the metric with respect to a unit area flat metric in its conformal class.

Higher genusEdit

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 alsoEdit

ReferencesEdit

  • Horowitz, Charles; Katz, Karin Usadi; Katz, Mikhail G. (2009). "Loewner's torus inequality with isosystolic defect". Journal of Geometric Analysis. 19 (4): 796–808. arXiv:0803.0690. doi:10.1007/s12220-009-9090-y. MR 2538936.
  • Katz, Mikhail G. (2007). Systolic geometry and topology. Mathematical Surveys and Monographs. 137. With an appendix by J. Solomon. Providence, RI: American Mathematical Society. doi:10.1090/surv/137. ISBN 978-0-8218-4177-8. MR 2292367.
  • Katz, Mikhail G.; Sabourau, Stéphane (2005). "Entropy of systolically extremal surfaces and asymptotic bounds". Ergodic Theory Dynam. Systems. 25 (4): 1209–1220. arXiv:math.DG/0410312. doi:10.1017/S0143385704001014. MR 2158402.
  • Katz, Mikhail G.; Sabourau, Stéphane (2006). "Hyperelliptic surfaces are Loewner". Proc. Amer. Math. Soc. 134 (4): 1189–1195. arXiv:math.DG/0407009. doi:10.1090/S0002-9939-05-08057-3. MR 2196056.
  • Pu, Pao Ming (1952). "Some inequalities in certain nonorientable Riemannian manifolds". Pacific J. Math. 2 (1): 55–71. MR 0048886.