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The main purpose of this paper is to propose a scheme of a proof of the nonsimpleness of the group ${\rm Homeo}^\Omega(D^2,\del D^2)$ of area preserving homeomorphisms of the 2-disc $D^2$. 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 $\Cal: {\rm Diff}^\Omega(D^1,\del D^2) \to \R$ to a homomorphism $\overline \Cal: {\rm Hameo(}D^2,\del D^2) \to \R$ to that of the vanishing of the basic phase function $f_{\underline{\mathbb F}}$, a Floer theoretic graph selector constructed in \cite{oh:jdg}, that is associated to the graph of the topological Hamiltonian loop and its normalized Hamiltonian $\underline{F}$ on $S^2$ that is obtained via the natural embedding $D^2 \hookrightarrow S^2$. Here ${\rm Hameo(}D^2,\del D^2)$ is the group of Hamiltonian homeomorphisms introduced by M\"uller and the author \cite{oh:hameo1}. We then provide an evidence of this vanishing conjecture by proving the conjecture for the special class of \emph{weakly graphical} topological Hamiltonian loops on $D^2$ via a study of the associated Hamiton-Jacobi equation.
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참고문헌 (21)
참고문헌 신청
A. Banyaga / 1978 / Sur la structure du groupe des diff´eomorphismes qui pr´eservent une forme symplectique / Comment. Math. Helvetici 53(2):174~227
L. Buhovsky / 2013 / Uniqueness of generating Hamiltonians for topolgical Hamiltonian flows / J. Symplectic Geom 11(1):37~52
E. Calabi / 1970 / Problmes in analysis / Princeton University Press :1~26
M. Chaperon / 1991 / Lois de conservation et g´eom´etrie symplectique / C. R. Acad. Sci. Paris S´er. I Math 312(4):345~348
Y. Eliashberg / 1987 / A theorem on the structure of wave fronts and its applications / (Russian)Funksional. Anal. i Prilzhen 21(3):65~72
L. Buhovsky / 2013 / Uniqueness of generating Hamiltonians for topolgical Hamiltonian flows / J. Symplectic Geom 11(1):37~52
E. Calabi / 1970 / Problmes in analysis / Princeton University Press :1~26
M. Chaperon / 1991 / Lois de conservation et g´eom´etrie symplectique / C. R. Acad. Sci. Paris S´er. I Math 312(4):345~348
Y. Eliashberg / 1987 / A theorem on the structure of wave fronts and its applications / (Russian)Funksional. Anal. i Prilzhen 21(3):65~72
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