Degenerate Maxima in Hamiltonian Systems

TL;DR

This paper investigates the properties of loops of Hamiltonian diffeomorphisms with degenerate fixed maxima, revealing restrictions on their degeneracy and implications for symplectic geometry and uniruled manifolds.

Contribution

It establishes that loops with degenerate fixed maxima cannot have totally degenerate fixed global maxima, providing new insights into Hamiltonian dynamics and symplectic topology.

Findings

Loops with degenerate fixed maxima cannot have totally degenerate fixed global maxima

Results impact the understanding of Hofer geometry in symplectic 4-manifolds

Provides criteria for certain 4-manifolds to be uniruled

Abstract

In this paper we explore loops of non-autonomous Hamiltonian diffeomorphisms with degenerate fixed maxima. We show that such loops can not have totally degenerate fixed global maxima. This has applications for the Hofer geometry of the group of Hamiltonians for certain symplectic 4 manifolds and also gives criteria for certain 4 manifolds to be uniruled.