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 Gromov (1981) 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 (Bangert et al. 2009).

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