Non-contractible orbits for Hamiltonian functions on Riemann surfaces

TL;DR

This paper proves the existence of non-contractible Hamiltonian orbits on Riemann surfaces under certain conditions, providing a new estimate for a symplectic capacity related to Hamiltonian dynamics.

Contribution

It introduces the first upper bound estimate for a generalized Biran-Polterovich-Salamon capacity on closed symplectic manifolds using non-contractible orbits.

Findings

Existence of non-contractible orbits under specific Hamiltonian conditions

First upper bound estimate for a generalized symplectic capacity

Application to Hamiltonian dynamics on Riemann surfaces

Abstract

We consider two disjoint and homotopic non-contractible embedded loops on a Riemann surface and prove the existence of a non-contractible orbit for a Hamiltonian function on the surface whenever it is sufficiently large on one of the loops and sufficiently small on the other one. This gives the first example of an estimate from above for a generalized form of the Biran-Polterovich-Salamon capacity for a closed symplectic manifold.