Systolic inequalities for K3 surfaces via stability conditions

We introduce the notions of categorical systoles and categorical volumes of Bridgeland stability conditions on triangulated categories. We prove that for any projective K3 surface X, there exists a constant C depending only on the rank and discriminant of NS(X), such that $$\begin{aligned} \mathrm {sys}(\sigma )^2\le C\cdot \mathrm {vol}(\sigma ) \end{aligned}$$ sys ( σ ) 2 ≤ C · vol ( σ ) holds for any stability condition on $$\mathcal {D}^b\mathrm {Coh}(X)$$ D b Coh ( X ) . This is an algebro-geometric generalization of a classical systolic inequality on two-tori. We also discuss applications of this inequality in symplectic geometry.

A systolic inequality with remainder in the real projective plane

The first paper in systolic geometry was published by Loewner’s student P. M. Pu over half a century ago. Pu proved an inequality relating the systole and the area of an arbitrary metric in the real projective plane. We prove a stronger version of Pu’s systolic inequality with a remainder term.

On metrics on 2-orbifolds all of whose geodesics are closed

We show that the periods and the topology of the space of closed geodesics on a Riemannian 2-orbifold all of whose geodesics are closed depend, up to scaling, only on the orbifold topology and compute it. In the manifold case we recover the fact proved by Gromoll, Grove and Pries that all prime geodesics have the same length, without referring to the existence of simple geodesics. We partly strengthen our result in terms of conjugacy of contact forms and explain how to deduce rigidity on the projective plane based on a systolic inequality due to Pu.

A systolic-like extremal genus two surface

It is known that the genus two surface admits a piecewise flat metric with conical singularities which is extremal for the systolic inequality among all nonpositively curved metrics. We prove that this piecewise flat metric is also critical for slow metric variations, without curvature restrictions, for another type of systolic inequality involving the lengths of the shortest noncontractible loops in different free homotopy classes.

Fiberwise convexity of Hill’s lunar problem

In this paper, we prove the fiberwise convexity of the regularized Hill’s lunar problem below the critical energy level. This allows us to see Hill’s lunar problem of any energy level below the critical value as the Legendre transformation of a geodesic problem on [Formula: see text] with a family of Finsler metrics. Therefore the compactified energy hypersurfaces below the critical energy level have the unique tight contact structure on [Formula: see text]. Also one can apply the systolic inequality of Finsler geometry to the regularized Hill’s lunar problem.

Optimal systolic inequalities on Finsler Möbius bands

We prove optimal systolic inequalities on Finsler Möbius bands relating the systole and the height of the Möbius band to its Holmes–Thompson volume. We also establish an optimal systolic inequality for Finsler Klein bottles of revolution, which we conjecture to hold true for arbitrary Finsler metrics. Extremal metric families both on the Möbius band and the Klein bottle are also presented.