Loewner's torus inequality

It relates the systole and the area of an arbitrary Riemannian metric on the 2-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 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

Geometrically, this torus can be obtained by gluing opposite pairs of edges of either a regular hexagon, or a rhombus with 60° and 120° angles.

is the Riemannian distance, namely least length of a path joining

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.

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.

Whether or not the inequality is satisfied by all surfaces of nonpositive Euler characteristic is unknown.