Systolic Geometry - Gromov's Stable Inequality

Gromov's Stable Inequality

A significant difference between 1-systolic invariants (defined in terms of lengths of loops) and the higher, k-systolic invariants (defined in terms of areas of cycles, etc.) should be kept in mind. While a number of optimal systolic inequalities, involving the 1-systoles, have by now been obtained, just about the only optimal inequality involving purely the higher k-systoles is Gromov's optimal stable 2-systolic inequality

for complex projective space, where the optimal bound is attained by the symmetric Fubini-Study metric, pointing to the link to quantum mechanics. Here the stable 2-systole of a Riemannian manifold M is defined by setting

where is the stable norm, while λ₁ is the least norm of a nonzero element of the lattice. Just how exceptional Gromov's stable inequality is, only became clear recently. Namely, it was discovered that, contrary to expectation, the symmetric metric on the quaternionic projective plane is not its systolically optimal metric, in contrast with the 2-systole in the complex case. While the quaternionic projective plane with its symmetric metric has a middle-dimensional stable systolic ratio of 10/3, the analogous ratio for the symmetric metric of the complex projective 4-space gives the value 6, while the best available upper bound for such a ratio of an arbitrary metric on both of these spaces is 14. This upper bound is related to properties of the Lie algebra E7. If there exists an 8-manifold with exceptional Spin(7) holonomy and 4-th Betti number 1, then the value 14 is in fact optimal. Manifolds with Spin(7) holonomy have been studied intensively by Dominic Joyce.