CONTINUOUS HAMILTONIAN DYNAMICS AND AREA-PRESERVING HOMEOMORPHISM GROUP OF D^{2}

- Journal title : Journal of the Korean Mathematical Society
- Volume 53, Issue 4, 2016, pp.795-834
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.j150288

Title & Authors

CONTINUOUS HAMILTONIAN DYNAMICS AND AREA-PRESERVING HOMEOMORPHISM GROUP OF D^{2}

Oh, Yong-Geun;

Oh, Yong-Geun;

Abstract

The main purpose of this paper is to propose a scheme of a proof of the nonsimpleness of the group (, ) of area preserving homeomorphisms of the 2-disc . We first establish the existence of Alexander isotopy in the category of Hamiltonian homeomorphisms. This reduces the question of extendability of the well-known Calabi homomorphism Cal : (, ) to a homomorphism : Hameo(, ) to that of the vanishing of the basic phase function , a Floer theoretic graph selector constructed in [9], that is associated to the graph of the topological Hamiltonian loop and its normalized Hamiltonian on that is obtained via the natural embedding . Here Hameo(, ) is the group of Hamiltonian homeomorphisms introduced by and the author [18]. We then provide an evidence of this vanishing conjecture by proving the conjecture for the special class of weakly graphical topological Hamiltonian loops on via a study of the associated Hamiton-Jacobi equation.

Keywords

area-preserving homeomorphism group;Calabi invariant;Lagrangian submanifolds;generating function;basic phase function;topological Hamiltonian loop;Hamilton-Jacobi equation;

Language

English

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