Systoles of surfaces

In mathematics, systolic inequalities for curves on surfaces were first studied by Charles Loewner in 1949 (unpublished; see remark at end of P. M. Pu's paper in '52).

In 1949 Loewner proved his inequality for metrics on the torus T2, namely that the systolic ratio SR(T2) is bounded above by

, with equality in the flat (constant curvature) case of the equilateral torus (see hexagonal lattice).

A similar result is given by 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(RP2), also attained in the constant curvature case.

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.

Namely, they exhibited arithmetic hyperbolic Riemann surfaces with systole behaving as a constant times

Note that area is 4π(g-1) from the Gauss-Bonnet theorem, so that SR(g) behaves asymptotically as a constant times

defined by a tower of principal congruence subgroups of the (2,3,7) hyperbolic triangle group satisfy the bound resulting from an analysis of the Hurwitz quaternion order.

This 2007 result by Mikhail Katz, Mary Schaps, and Uzi Vishne improves an inequality due to Peter Buser and Peter Sarnak in the case of arithmetic groups defined over

For the Hurwitz surfaces of principal congruence type, the systolic ratio SR(g) is asymptotic to Using Katok's entropy inequality, the following asymptotic upper bound for SR(g) was found in (Katz-Sabourau 2005): see also (Katz 2007), p. 85.

Combining the two estimates, one obtains tight bounds for the asymptotic behavior of the systolic ratio of surfaces.

A lower bound of 1/961 obtained by Croke in '88 has recently been improved by Nabutovsky, Rotman, and Sabourau.