Loewner's torus inequality

Remarks

By the uniformization theorem, $g$ is conformal to a flat metric $g_0$, so $g = f^2 g_0$ for a positive function $f$. Averaging $f$ over the translations of the flat torus gives a flat metric. By the Cauchy–Schwarz inequality its area is at most that of $g$, and by averaging lengths over translates of a closed geodesic its systole is at least that of $g$. For a flat torus $\R^2 / \Lambda$, the claim is that the shortest nonzero vector $v$ of a lattice $\Lambda$ of covolume $A$ satisfies $\abs{v}^2 \le \tfrac{2}{\sqrt{3}} A$. This follows by reduction of a basis of $\Lambda$, and the hexagonal lattice is the extreme case.

For the square torus $\R^2/\Z^2$ the two sides are $1$ and $2/\sqrt{3} \approx 1.155$. For the hexagonal torus above, the systole is $1$ and the area is $\sqrt{3}/2$, so equality holds.

Going deeper

For a student

This is the first result of systolic geometry, and the model for Gromov's systolic inequality in higher dimensions. The constant $2/\sqrt{3}$ is the Hermite constant $\gamma_2$ of the geometry of numbers. The proof pattern, reduction to the most symmetric metrics in a conformal class by averaging, recurs in other optimal systolic inequalities.