Lagrangian configurations and Hamiltonian maps

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. LEONID POLTEROVICH AND EGOR SHELUKHIN 7. Lagrangian spectral invariants and estimators 23 7.1. Lagrangian Floer homology with bounding cochains and bulk deformation 23 7.2. Lagrangian spectral invariants 26 7.3. Orbifold setting 28 8. Further directions 39 8.1. Other configurations 39 8.2. Next destination: asymptotic cone of Ham(S 2 ) 40 8.3. Comparison with periodic Floer Homology? 41 Acknowledgements 41 References 41

On the Hofer-Zehnder conjecture on weighted projective spaces

“…Major works in symplectic topology recently used symplectic orbifolds [7,26] and we are hoping that this study will contribute to a better understanding of what one should expect of an orbifold Hamiltonian Floer homology theory. Indeed, our "weighted" result is not the first intriguing phenomenon observed in this topic: a recent work extending the Floer homology to global quotient orbifolds (i.e.…”

On two remarkable groups of area-preserving homeomorphisms

“…Recently, a number of breakthrough works addressed the Fathi question, where it was first solved in the case of a two-disc [1], then in the two-sphere case [3,9], and finally for general surfaces of finite type and finite area [2]. Moreover, the works [3,9] have largely contributed to Hofer geometry, in particular solving the Polterovich-Kapovich question. Powerful tools coming from Floer homology, embedded contact homology and periodic homology theories, were central in making that progress possible.…”

“…The aim of the present article is to give an additional insight on that picture. Our approach is based on application of novel Floer-theoretic invariants from [9] (see also [6]), in combination with a soft approach relying on an idea due to Sikorav [10].…”

“…It is well known [8,2] that Hameo(M, ω) ⊂ FHomeo(M, ω) are normal subgroups of Ham(M, ω). For symplectic surfaces (M, ω) of finite type and area it was shown [1,3,9,2] that FHomeo(M, ω) is in fact a proper normal subgroup of Ham(M, ω). This answers the Fathi question to the negative.…”

Exact orbifold fillings of contact manifolds

“…For example, Mak-Smith [MS21] used orbifold Lagrangian Floer theory developed by Cho-Poddar [CP14] to obtain new families of non-displaceable Lagrangian links in symplectic four-manifolds. The idea was further exploited by Polterovich-Shelukhin [PS21] as well as Cristofaro-Gardiner-Humilière-Mak-Seyfaddini-Smith [CGHM + 21] leading to recent breakthroughs on dynamics on surfaces and C 0 symplectic geometry.…”