THE GRADIENT FLOW EQUATION OF RABINOWITZ ACTION FUNCTIONAL IN A SYMPLECTIZATION
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Rabinowitz action functional is the Lagrange multiplier functional of the negative area functional to a constraint given by the mean value of a Hamiltonian. In this note we show that on a symplectization there is a one-to-one correspondence between gradient flow lines of Rabinowitz action functional and gradient flow lines of the restriction of the negative area functional to the constraint. In the appendix we explain the motivation behind this result. Namely that the restricted functional satisfies Chas-Sullivan additivity for concatenation of loops which the Rabinowitz action functional does in general not do.
Abstract
Rabinowitz action functional is the Lagrange multiplier functional of the negative area functional to a constraint given by the mean value of a Hamiltonian. In this note we show that on a symplectization there is a one-to-one correspondence between gradient flow lines of Rabinowitz action functional and gradient flow lines of the restriction of the negative area functional to the constraint. In the appendix we explain the motivation behind this result. Namely that the restricted functional satisfies Chas-Sullivan additivity for concatenation of loops which the Rabinowitz action functional does in general not do.
참고문헌 (19)
- A. Abbondandolo, "Floer homology on the time-energy extended phase space", J. Symplectic Geom, 16(2), pp.279~355 (2018)[DOI]
- P. Albers, "Symplectic Tate homology", Proc. Lond. Math. Soc, 112(1), pp.169~205 (2016)[DOI]
- L. Bieberbach, "Δu = eu und die automorphen Funktionen", Math. Ann, 77(2), pp.173~212 (1916)[DOI]
- M. Chas, "String topology", ~ (1999)
- K. Cieliebak, "Symplectic homology. II. A general construction", Math. Z, 218(1), pp.103~122 (1995)[DOI]
- K. Cieliebak, "A Floer homology for exact contact embeddings", Pacific J. Math, 239(2), pp.251~316 (2009)[DOI]
- K. Cieliebak, "Rabinowitz Floer homology and symplectic homology", Ann. Sci. Ec. Norm. Super, 43(6), pp.957~1015 (2010)[DOI]
- K. Cieliebak, "Symplectic homology and the Eilenberg-Steenrod axioms", Algebr. Geom. Topol, 18(4), pp.1953~2130 (2018)[DOI]
- S. Dostoglou, "Self-dual instantons and holomorphic curves", Ann. of Math, 139(3), pp.581~640 (1994)[DOI]
- Y. Eliashberg, "Introduction to symplectic field theory", Geom. Funct. Anal, pp.560~673 (2000)[DOI]
- R. P. Feynman, "Space-time approach to non-relativistic quantum mechanics", Rev. Modern Physics, 20, pp.367~387 (1948)[DOI]
- U. Frauenfelder, "Vortices on the cylinder", Int. Math. Res. Not, 2006, pp.34~ (2006)[DOI]
- A. R. P. Gaio, "Gromov-Witten invariants of symplectic quotients and adiabatic limits", J. Symplectic Geom, 3(1), pp.55~159 (2005)[DOI]
- N. Hingston, "Loop products, Poincar´e duality, index growth and dynamics", Proceedings of the International Congress of Mathematicians-Seoul 2014, 2, pp.881~895 (2014), Kyung Moon Sa
- J. L. Kazdan, "Curvature functions for compact 2-manifolds", Ann. of Math, 99(2), pp.14~47 (1974)[DOI]
- W. Rudin, "Functional Analysis" (1973), McGraw-Hill Book Co
- D. A. Salamon, "Floer homology and the heat flow", Geom. Funct. Anal, 16(5), pp.1050~1138 (2006)[DOI]
- E. Stueckelberg, "La signification du temps propre en m´ecanique ondulatoire", Helv. Phys. Acta, 14, pp.322~323 (1941)
- C. Viterbo, "Functors and computations in Floer homology with applications I", Geom. Funct. Anal, 9(5), pp.985~1033 (1999)[DOI]