CONTINUOUS HAMILTONIAN DYNAMICS AND AREA-PRESERVING HOMEOMORPHISM GROUP OF D2

Title & Authors
CONTINUOUS HAMILTONIAN DYNAMICS AND AREA-PRESERVING HOMEOMORPHISM GROUP OF D2
Oh, Yong-Geun;

Abstract
The main purpose of this paper is to propose a scheme of a proof of the nonsimpleness of the group $\small{Homeo^{\Omega}}$ ($\small{D^2}$, $\small{{\partial}D^2}$) of area preserving homeomorphisms of the 2-disc $\small{D^2}$. 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 : $\small{Diff^{\Omega}}$ ($\small{D^1}$, $\small{{\partial}D^2}$)$\small{{\rightarrow}{\mathbb{R}}}$ to a homomorphism $\small{{\bar{Cal}}}$ : Hameo($\small{D^2}$, $\small{{\partial}D^2}$)$\small{{\rightarrow}{\mathbb{R}}}$ to that of the vanishing of the basic phase function $\small{f_{\underline{F}}}$, a Floer theoretic graph selector constructed in [9], that is associated to the graph of the topological Hamiltonian loop and its normalized Hamiltonian $\small{{\underline{F}}}$ on $\small{S^2}$ that is obtained via the natural embedding $\small{D^2{\hookrightarrow}S^2}$. Here Hameo($\small{D^2}$, $\small{{\partial}D^2}$) is the group of Hamiltonian homeomorphisms introduced by $\small{M{\ddot{u}}ller}$ 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 $\small{D^2}$ 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
Cited by
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