Lagrangian configurations and Hamiltonian maps

2021/02/11 by Leonid Polterovich, Polterovich, Leonid, Egor Shelukhin +1 · 7 citations
Mathematics · #53Dxx #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2102.06118

openalex publication_date 2021/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamiltonian group of the two-sphere equipped with Hofer's metric, prove constraints on Lagrangian packing, find instances of Lagrangian Poincaré recurrence, and present a new hierarchy of normal subgroups of area-preserving homeomorphisms of the two-sphere. The technology involves Lagrangian spectral invariants with Hamiltonian term in symmetric product orbifolds.

Cited by

• Legendrian barriers in prequantization bundles
• Symplectic Weyl Laws
• Orbifold Floer spectral invariants, symmetric product links and Weyl laws
• Balanced curves, quasimorphisms, and the equator conjecture
• The L^p-diameter of the space of contractible loops
• Billiards and Hofer's Geometry
• The infinite dimensional geometry of conjugation invariant generating sets

Related

• On the Lagrangian Hofer geometry in symplectically aspherical manifolds
• A Comparison of Hofer's Metrics on Hamiltonian Diffeomorphisms and Lagrangian Submanifolds
• Loops of Lagrangian submanifolds and pseudoholomorphic discs
• Lagrangian metric geometry with Riemannian bounds
• Lagrangian Non-Intersections