Optimal systolic inequalities

The list of optimal systolic inequalities appearing below is surely incomplete. Please signal omissions to katzmik (at) macs.removethis.biu.andthis.ac.il

Optimal inequalities for closed surfaces:

1. Loewner inequality for the torus ('49)
2. Pu's inequality for the real projective plane ('52)
3. Rodin's optimal inequality for closed surfaces ('68)
4. Bavard's inequality for the Klein bottle ('86)
5. In genus 2: F. Jenni , C. Bavard, P. Schmutz in hyperbolic case. S. Sabourau and M. Katz in CAT(0) case
6. For relative systole of the quotient of a hyperelliptic surface by a fixed point free, real involution, by V. Bangert, C. Croke, S. Ivanov, and M. Katz
7. Relative systole of genus 2 surface, in hyperbolic case:
H. Parlier

Optimal inequalities for manifolds of dimension 3 or more:

9. Gromov's inequality for the stable 2-systole of complex projective space
10. Burago-Ivanov-Gromov inequality for the stable 1-systole of the n-torus (or more generally, n-manifold with real cup-length n), in terms of the n-th Hermite constant
11. An optimal (1,n-1) inequality, in terms of the n-th Bergé-Martinet constant, by Bangert and Katz
12. inequality combining 1-systole and stable 1-systole, in terms of the Hermite constant, generalizing item 10, by Ivanov and Katz
13. Inequality of item 12 is valid for 3-manifolds with b1=2 and with non-vanishing Lescop invariant, by Katz and Lescop based on argument of Marin
14. Inequality combining 1-systole and conformal 1-systole, generalizing the above, by Bangert, Croke, Ivanov, Katz

Most of the above inequalities are discussed in the recent monograph Systolic geometry and topology.

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