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In mathematics, systolic inequalities for curves on surfaces were first studied by Charles Loewner in 1949 (unpublished; see remark at end of P. M. Pu's paper in '52). Given a closed surface, its systole, denoted sys, is defined to be 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/sys2. The systolic ratio SR is the reciprocal quantity sys2/area. See also Introduction to systolic geometry. 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 the Klein bottle K, Bavard (1986) obtained an optimal upper bound of for the systolic ratio: based on work by Blatter from the 1960s. An orientable surface of genus 2 satisfies Loewner's bound , 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). 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 found an upper bound for SR(g) given by a constant times 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 . Note that area is 4π(g-1) from the Gauss-Bonnet theorem, so that SR(g) behaves asymptotically as a constant times . The study of the asymptotic behavior for large genus of the systole of hyperbolic surfaces reveals some interesting constants.