Elementary spectral invariants and three-dimensional Reeb dynamics

TL;DR

This paper reviews recent advances in understanding periodic Reeb orbits in three-dimensional contact manifolds, focusing on elementary spectral invariants and their relation to embedded contact homology and symplectic capacities.

Contribution

It introduces elementary spectral invariants as simplified tools for studying Reeb dynamics, connecting them to ECH capacities and providing new insights into periodic orbit existence.

Findings

Elementary spectral invariants simplify analysis of Reeb orbits.

Connections established between spectral invariants and ECH capacities.

Survey of recent results on periodic Reeb orbits in 3D contact manifolds.

Abstract

We survey various recent results on the existence and properties of periodic orbits of Reeb vector fields in three dimensions. We give an introduction to the "elementary spectral invariants" of contact three-manifolds, and we explain how they can be used to prove some of these results. (The remaining results can be proved using spectral invariants from embedded contact homology, of which the elementary spectral invariants are a simplification.) We then review the "alternative ECH capacities" of symplectic four-manifolds, and explain how these can be modified to define the elementary spectral invariants.