Burton Rodin

See item 5 below for a prize.

1. The first systolic inequality for surfaces with a constant independent of the genus appeared implicitly in the '68 paper by B. Rodin, see bibliography (pdf). The inequality in question is the bound

sys2 ≤ 2 area,

improving earlier genus-dependent bounds by R. Accola and C. Blatter. The inequality results from Rodin's optimal inequality

g g* ≤ area(R),

which appears on page 372, line 5 of Rodin's paper. Here g is the length of a nonseparating loop c on a closed surface R, while g* is the distance between the two boundary components of the surface with boundary obtained by compactifying the complement R \ c. Applying this inequality to a systolic loop, together with a triangle inequality argument showing that g ≤ 2g* in this case, yields the bound sys2 ≤ 2 area.

2. The inequality sys2 ≤ 2 area, first appeared explicitly in the '80 book by Yu. Burago and V. Zalgaller, and the '82 paper by J. Hebda, see bibliography (pdf).

3. The inequality was improved by M. Gromov in '83 to the bound sys2 ≤ 43 area, for an arbitrary aspherical surface.

4. One expects Loewner's torus inequality,

sys2 ≤  23  area,

to be satisfied by all aspherical surfaces. However, this has only been verified for oriented surfaces of genus 2 as well as of genus 20 or above, in the work of M. Katz and S. Sabourau, see bibliography (pdf). The genera 3,4,5,...,19 are open.

5. Rodin prize. In the hallowed tradition of P. Erdös, we are hereby offering a prize of $50 for the determination of the asymptotic behavior of the systole of a hyperbolic surface when the genus becomes unbounded. For European contestants only, the prize is upped to €50.

More precisely, the largest systole of a hyperbolic surface of genus g is known to behave asymptotically as

C log(g).

A combination of the work, on the one hand, of Katz and Sabourau, and on the other, of Katz, Schaps, and Vishne, shows that the constant C must be contained in the interval [43, 3]. Here the lower bound is known at least for a suitable sequence of genera, while the asymptotic upper bound is in fact valid for an arbitrary Riemannian metric of area normalized to the value 2π(2g-2). The problem is to determine the precise asymptotic behavior in g of the maximal systole in genus g. Weigh also other prizes.

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