Pu's inequality

In the 1950's, Charles Loewner's student P. M. Pu [Pu52] proved the following theorem. Let  ${\mathbb{R}\mathbb{P}}^2$ be the real projective plane endowed with an arbitrary metric, i.e. an imbedding in some  ${\mathbb{R}}^n$. Then

$$\left(\frac{L}{\pi} \right)^2\leq \frac{A}{2\pi},$$ (2.1)

where $A$ is its total area and $L$ is the least length of a non-contractible loop. This isosystolic inequality, or simply systolic inequality for short, is also sharp, to the extent that equality is attained precisely for a metric of constant Gaussian curvature, namely antipodal quotient of a round sphere. In the systolic notation where $L$ is replaced by $\pisys _1$, Pu's inequality takes the following form:

$$\pisys _1({\mathcal G})^2 \leq \tfrac{\pi}{2} \area ({\mathcal G}),$$ (2.2)

for every metric  ${\mathcal G}$ on  ${\mathbb{R}\mathbb{P}}^2$.

For a proof, see [Ka07, Section 6.5].

Pu's inequality can be generalized as follows.

Theorem 2.1   Every surface  $(\Sigma ,{\mathcal G})$ different from $S^2$ satisfies the optimal bound (2.2), attained precisely when, on the one hand, the surface $\Sigma$ is a real projective plane, and on the other, the metric  ${\mathcal G}$ is of constant Gaussian curvature.

The extension to surfaces of nonpositive Euler characteristic follows from Gromov's inequality (2.3) below (by comparing the numerical values of the two constants). Namely, every aspherical compact surface  $(\Sigma ,{\mathcal G})$ admits a metric ball

$$B=B_p\left(\tfrac{1}{2}\pisys _1({\mathcal G})\right) \subset \Sigma$$

of radius  $\tfrac{1}{2}\pisys _1({\mathcal G})$ which satisfies [Gro83, Corollary 5.2.B]

$$\pisys _1({\mathcal G})^2 \leq \frac{4}{3}\area (B).$$ (2.3)