Contact quasi-states and their applications

TL;DR

This work constructs partial contact quasi-states $\\zeta_\\alpha$ and contact quasi-measures $\\tau_\\alpha$ on Liouville-fillable contact manifolds by leveraging a contact spectral invariant from $HF_*$ and a nontrivial symplectic homology of the filling. The approach parallels Entov-Polterovich’s symplectic framework, with additional nuance from contact geometry manifested as error terms in general (non-Reeb-invariant) settings. The authors establish existence and core axioms for these objects, derive a way to obtain a subadditive quasi-measure from the quasi-state, and prove a rigidity result leading to a (new) proof of the contact big fibre theorem under the assumption that the filling’s symplectic homology is nonzero and $\mathbb{Z}$-graded. Overall, the paper strengthens the contact-geometry toolbox for rigidity phenomena and provides a classical-leaning route to a key theorem in the field under a precise topological condition.

Abstract

We introduce the notions of partial contact quasi-state and contact quasi-measure. Using the contact spectral invariant from the work by Djordjević-Uljarević-Zhang, one can construct partial contact quasi-states and contact quasi-measures on each contact manifold fillable by a Liouville domain with non-vanishing and $\mathbb{Z}$-graded symplectic homology. As an application, we present an alternative proof of the contact big fibre theorem from the recent work by Sun-Uljarević-Varolgunes under a mild topological condition. Our proof follows a more "classical" approach developed by Entov-Polterovich in the symplectic setting.

Theorems & Definitions (22)

• Theorem 1.1: cf. Corollary 1.10 in SUV25
• Theorem 2.1: DUZ23
• Proposition 2.2: DUZ23
• Proposition 2.3
• Definition 3.1
• Theorem 3.2
• Definition 3.3
• Corollary 3.4
• proof : Proof of Proposition \ref{['prop:conj']}
• Definition 4.1