Gromov's inequality for complex projective space

In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality


 * $$\mathrm{stsys}_2{}^n \leq n!

\;\mathrm{vol}_{2n}(\mathbb{CP}^n)$$,

valid for an arbitrary Riemannian metric on the complex projective space, where the optimal bound is attained by the symmetric Fubini–Study metric, providing a natural geometrisation of quantum mechanics. Here $$\operatorname{stsys_2}$$ is the stable 2-systole, which in this case can be defined as the infimum of the areas of rational 2-cycles representing the class of the complex projective line $$\mathbb{CP}^1 \subset \mathbb{CP}^n$$ in 2-dimensional homology.

The inequality first appeared in as Theorem 4.36.

The proof of Gromov's inequality relies on the Wirtinger inequality for exterior 2-forms.

Projective planes over division algebras $$ \mathbb{R,C,H}$$
In the special case n=2, Gromov's inequality becomes $$\mathrm{stsys}_2{}^2 \leq 2 \mathrm{vol}_4(\mathbb{CP}^2)$$. This inequality can be thought of as an analog of Pu's inequality for the real projective plane $$\mathbb{RP}^2$$. In both cases, the boundary case of equality is attained by the symmetric metric of the projective plane. Meanwhile, in the quaternionic case, the symmetric metric on $$\mathbb{HP}^2$$ is not its systolically optimal metric. In other words, the manifold $$\mathbb{HP}^2$$ admits Riemannian metrics with higher systolic ratio $$\mathrm{stsys}_4{}^2/\mathrm{vol}_8$$ than for its symmetric metric.