Graphicality, C0 convergence, and the Calabi homomorphism
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Consider a sequence of compactly supported Hamiltonian diffeomorphisms $\phi_k$ of an exact symplectic manifold, all of which are ``graphical'' in the sense that their graphs are identified by a Darboux-Weinstein chart with the image of a one-form. We show by an elementary argument that if the $\phi_k$ $C^0$-converge to the identity, then their Calabi invariants converge to zero. This generalizes a result of Oh, in which the ambient manifold was the two-disk and an additional assumption was made on the Hamiltonians generating the $\phi_k$. We discuss connections to the open problem of whether the Calabi homomorphism extends to the Hamiltonian homeomorphism group. The proof is based on a relationship between the Calabi invariant of a $C^0$-small Hamiltonian diffeomorphism and the generalized phase function of its graph.
Abstract
Consider a sequence of compactly supported Hamiltonian diffeomorphisms $\phi_k$ of an exact symplectic manifold, all of which are ``graphical'' in the sense that their graphs are identified by a Darboux-Weinstein chart with the image of a one-form. We show by an elementary argument that if the $\phi_k$ $C^0$-converge to the identity, then their Calabi invariants converge to zero. This generalizes a result of Oh, in which the ambient manifold was the two-disk and an additional assumption was made on the Hamiltonians generating the $\phi_k$. We discuss connections to the open problem of whether the Calabi homomorphism extends to the Hamiltonian homeomorphism group. The proof is based on a relationship between the Calabi invariant of a $C^0$-small Hamiltonian diffeomorphism and the generalized phase function of its graph.
참고문헌 (10)
- A. Banyaga, "Sur la structure du groupe des diffeomorphismes qui preservent une forme symplectique", Comment. Math. Helv, 53(2), pp.174~227 (1978)
- S. Bates, "Lectures on the Geometry of Quantization", Berkeley Mathematics Lecture Notes, 8 (1997), AMS, Berkeley Center for Pure and Applied Mathematics
- L. Buhovsky, "Uniqueness of generating Hamiltonians for topological Hamiltonian ows", J. Symplectic Geom, 11(1), pp.37~52 (2013)
- E. Calabi, "On the group of automorphisms of a symplectic manifold", Problems in analysis (Lectures at the Sympos. in honor of Salomon Bochner, Princeton Univ., Princeton, N.J., 1969), pp.1~26 (1970), Princeton Univ. Press
- Y. Eliashberg, "A theorem on the structure of wave fronts and its application in sym-plectic topology", Funktsional. Anal. i Prilozhen, 21(3), pp.65~72 (1987)
- Y.-G. Oh, "The group of Hamiltonian homeomorphisms and continuous Hamiltonian ows, In Symplectic topology and measure preserving dynamical systems", Contemp. Math., 512, pp.149~177 (2010), Amer. Math. Soc.
- 오용근, "Continuous Hamiltonian dynamics and area-preserving homeomorphism group of $D^2$", 대한수학회지, 53(4), pp.795~834 (2016), 대한수학회
- Y.-G. Oh, "The group of Hamiltonian homeomorphisms and C0-symplectic topology", J. Symplectic Geom, 5(2), pp.167~219 (2007)
- J.-C. Sikorav, "Problemes d'intersections et de points xes en geometrie hamiltonienne", Comment. Math. Helv, 62(1), pp.62~73 (1987)
- C. Viterbo, "On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians ows", Int. Math. Res. Not, 2006, pp.9~ (2006)