# Two closed orbits for non-degenerate Reeb flows

## TL;DR

This paper proves the existence of at least two distinct closed Reeb orbits on certain contact manifolds with symplectic fillings, using equivariant symplectic homology, and extends results to various examples and operations.

## Contribution

It establishes new conditions under which multiple closed Reeb orbits exist, linking symplectic homology to Reeb dynamics on a broad class of contact manifolds.

## Key findings

- At least two geometrically distinct closed Reeb orbits exist under specified conditions.
- The equivariant symplectic homology condition is preserved under boundary connected sums.
- Application to a Lusternik-Fet type theorem for Reeb flows on certain manifolds.

## Abstract

We prove that every non-degenerate Reeb flow on a closed contact manifold $M$ admitting a strong symplectic filling $W$ with vanishing first Chern class carries at least two geometrically distinct closed orbits provided that the positive equivariant symplectic homology of $W$ satisfies a mild condition. Under further assumptions, we establish the existence of two geometrically distinct closed orbits on any contact finite quotient of $M$. Several examples of such contact manifolds are provided, like displaceable ones, unit cosphere bundles, prequantization circle bundles, Brieskorn spheres and toric contact manifolds. We also show that this condition on the equivariant symplectic homology is preserved by boundary connected sums of Liouville domains. As a byproduct of one of our applications, we prove a sort of Lusternik-Fet theorem for Reeb flows on the unit cosphere bundle of not rationally aspherical manifolds satisfying suitable additional assumptions.