Invariant contact structures and systolic inequalities

• Applicant: Dr. Simon Vialaret
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 572978773

A contact manifold is a manifold endowed with a contact form. It carries a canonical flow, the Reeb flow of the contact form. A class of contact form on a contact manifold of dimension 2n-1 satisfies a systolic inequality when there is a uniform upper bound in this class on the systolic ratio, namely the ratio between the n-th power of the smallest period of a closed orbit of the Reeb flow, and the contact volume of the manifold. This generalizes a well-studied notion from Riemannian geometry. Moreover, it is known that a conjecture of Viterbo on symplectic capacities of convex domains, and a conjecture of Mahler on the volume product of convex sets, can be restated as systolic inequalities for certain contact forms on odd-dimensional spheres. It is known that, given any closed manifold, the set of all contact forms on that manifold never satisfies a systolic inequality. On the other hand, the local maximizers of the systolic ratio are fully characterized. This interplay between flexible (non-boundedness of the systolic ratio) and rigid (fully characterized local maximizers, boundedness of the systolic ratio in presence of convexity assumption) phenomena is characteristic of contact geometry. In my PhD thesis, I explored some aspects of this dichotomy. I showed that, on three-dimensional Seifert bundles, the class of contact forms invariant under the underlying circle action satisfies a systolic inequality, under the assumption that the Seifert space has non-zero Euler class. In the case of tight contact forms on principal bundles over the two-sphere, I obtained the optimal bound on the systolic ratio, and described saturating contact forms. This result is essentially sharp: the systolic inequality does not hold anymore for contact forms unvariant under a discrete group action. In this project, I want to continue the study of systolic inequalities for contact forms invariant under group actions. It has two main directions. First, I plan on focusing in the higher-dimensional setting. In that case, the diversity of possible group actions is much broader. I expect both systolic freedom and systolic inequalities to occur for invariant contact forms, depending on the dimension of the group acting. Moreover, I would like to prove sharp inequalities for specific classes of contact forms, namely cohomogeneity-one Riemannian metrics, and starshaped domains in the standard symplectic space invariant under a torus action. The second direction is to prove sharp systolic inequalities for contact forms on the three-torus, that are invariant under a two-torus action. This class of Reeb flows is unexpectedly rich, and a systolic inequality in that setting can be translated in a number geometric statement about the existence of integer points in the interior of certain starshaped domains in the plane. I plan to study the relation between such number-geometric result, systolic inequalities, and symplectic capacities of certain Lagrangian products in R4.

• DFG Programme: WBP Position