Loewner's Torus Inequality
Similarly to Pu's inequality, Loewner's torus inequality relates the total area, to the systole, i.e. least length of a noncontractible loop on the torus :
The boundary case of equality is attained if and only if the metric is homothetic to the flat metric obtained as the quotient of by the lattice formed by the Eisenstein integers.
Read more about this topic: Introduction To Systolic Geometry
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