Elementary spectral invariants and quantitative closing lemmas for contact three-manifolds

TL;DR

This paper introduces an elementary alternative to ECH spectra for contact three-manifolds, providing new tools for Reeb dynamics and establishing sharp quantitative closing lemmas, especially for irrational ellipsoids.

Contribution

It develops a new spectrum for contact three-manifolds avoiding Seiberg-Witten theory, enabling quantitative Reeb dynamics analysis and closing lemmas.

Findings

Defined an alternative to ECH spectrum using holomorphic curves.

Applied the spectrum to obtain quantitative closing lemmas.

Achieved a sharp closing lemma for irrational ellipsoids.

Abstract

In a previous paper, we defined an "elementary" alternative to the ECH capacities of symplectic four-manifolds, using max-min energy of holomorphic curves subject to point constraints, and avoiding the use of Seiberg-Witten theory. In the present paper we use a variant of this construction to define an alternative to the ECH spectrum of a contact three-manifold. The alternative spectrum has applications to Reeb dynamics in three dimensions. In particular, we adapt ideas from a previous joint paper with Edtmair to obtain quantitative closing lemmas for Reeb vector fields in three dimensions. For the example of an irrational ellipsoid, we obtain a sharp quantitative closing lemma.