Contact topology and hydrodynamics: I. Beltrami fields and the Seifert conjecture

Nonlinearity

Volumn 13, Issue 2, 2000, Pages 441-458

  Etnyre, John ^a   Ghrist, Robert ^b  

^a STANFORD UNIVERSITY   (United States)

^b GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0040213360     PISSN: 09517715     EISSN: None     Source Type: Journal    
DOI: 10.1088/0951-7715/13/2/306     Document Type: Article

References (62)

1. Abraham R and Marsden J 1978 Foundations of Mechanics (New York: Benjamin-Cummings)
2. Abraham R, Marsden J and Ratiu T 1988 Manifolds, Tensor Analysis and Applications (Applied Mathematics and Science vol 75) (Berlin: Springer)
3. Aebisher B 1994 Symplectic Geometry (Progress in Mathematics vol 124) (Berlin: Birkhäuser)
4. Alber H 1992 Existence of three-dimensional, steady, inviscid, incompressible flows with nonvanishing vorticity Math. Ann. 292 493-528
5. Claude Albert 1989 Le théorème de réduction de Marsden-Weinstein en géométrie cosympleclique et de contact J. Geom. Phys. 6 627-49
6. Arnold V I 1966 Sur la géométrie differentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits Ann. Inst. Fourier 16 316-61
7. Arnold V I 986 The asymptotic Hopf invariant and its applications Sel. Math. Sov. 5 327-45
8. Arnold V I and Khesin B 1992 Topological methods in hydrodynamics Ann. Rev. Fuid Mech. 24 145-66
9. Arnold V I and Khesin B 1998 Topological Methods in Hydrodynamics (Berlin: Springer)
10. Bennequin D 1983 Entrelacements et équations de Pfaff Asterisque 107-8 87-161
11. Bott R 1980 An equivariant setting of the Morse theory Enseign. Math. 26 271-8
12. Casasayas J. Martinez Alfaro J and Nunes A 1993 Knotted periodic orbits and integrability Hamillonian Systems and Celestial Mechanics (Guanajuato 1991) (Adv. Ser. Nonlinear Dynam. no 4) (River Edge, NJ: World Scientific ) pp 35-44
13. Constantin P and Majda A 1988 The Beltrami spectrum for incompressible fluid flows Commun. Math. Phys. 115 435-56
14. Dombre T, Frisch U, Greene J, Hénon M, Mehr A and Soward A 1986 Chaotic streamlines in the A B C flows J. Fluid Mech. 167 353-91
15. Eliashberg Y 1989 Classification of overtwisted contact structures on 3-manifolds Invent. Math. 98 623-37
16. Eliashberg Y 1990 Topological characterization of Stein manifolds of dimension >2 Int. J. Math. 1 29-46
17. Eliashberg Y 1992 Contact 3-manifolds twenty years since J Martinet's work Ann. Inst. Fourier 42 165-92
18. Eliashberg Y and Thurston W 1998 Confoliations University Lecture Series vol 13 (Providence, RI: American Mathematical Society)
19. Etnyre J 1997 Tight contact structures on lens spaces Preprint
20. Etnyre J and Ghrist R 1999 Contact topology and hydrodynamics II: solid tori Preprint
21. Etnyre J and Ghrist R 1999 Contact topology and hydrodynamics III: knotted flowlines Preprint to appear in Trans. Am. Math. Soc.
22. Etnyre J and Ghrist R 1999 Stratified integrals and unknots in inviscid flows Cont. Math. 246 99-112
23. Fomenko A T 1988 Integrability and Nonintegrability in Geometry and Mechanics (Dordrecht: Kluwer) (translated from the Russian by M V Tsaplina)
24. Fomenko A T 1991 Advances in Soviet Mathematics vol 6 (Providence. RI: American Mathematical Society)
25. Fomenko A T and Nguyen T Z 1991 Topological classification of integrable nondegenerate Hamiltonians on isoenergy three-dimensional spheres Advances in Soviet Mathematics vol 6, ed A T Fomenko (Providence, RI: American Mathematical Society) pp 267-96
26. Friedlander S, Gilbert A and Vishik M 1993 Hydrodynamic instability for certain A B C flows Geophys. Astrophys. Fluid Dynam. 73 97-107
27. Friedlander S, Strauss W and Vishik M 1997 Nonlinear instability in an ideal fluid Ann. Inst. H. Poincaré Anal. Non Linéaire 14 187-209
28. Friedlander S and Vishik M 1992 Instability criteria for steady flows of a perfect fluid Chaos 3 455-60
29. Frisch U and Orszag S 1990 Turbulence: challenges for theory and experiment Phys. Today 43 24-32
30. Ghrist R, Holmes P and Sullivan M 1997 Knots and Links in Three-Dimensional Flows (Springer Lecture Notes in Mathematics vol 1654) (Berlin: Springer)
31. Ginzburg V and Khesin B 1994 Steady fluid flows and symplectic geometry Geom. J. Phys. 14 195-210
32. Giroux E 1994 Une structure de contact, même tendue est plus ou moins tordue Ann. Sci. Ec. Norm. Sup. 27 697-705
33. Gompf R 1998 Handlebody construction of Stein surfaces Ann. Math. 148 619-93
34. Goresky M and MacPherson R 1988 Stratified Morse Theory (Berlin: Springer)
35. Gromov M 1985 Pseudoholomorphic curves in symplectic manifolds Invent. Math. 82 307-47
36. Haller G and Mezić I 1998 Reduction of three-dimensional volume-preserving flows with symmetry Nonlinearity 11 319-39
37. 2 counterexamples to the Seifert conjecture Topology 27 249-78
38. Hofer H 1993 Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three Invent. Math. 114 515-63
39. Hofer H and Kriener M 1997 Holomorphic curves in contact dynamics Notes from Park City Mathematics Institute (July 1997)
40. Hofer H, Wysocki K and Zender E 1996 A characterisation of the tight three-sphere Duke J. Math. 81 159-226
41. Hofer H and Zehnder E 1994 Hamiltonian Dynamics and Symplectic Invariants (Berlin: Birkhäuser)
42. Kanda Y 1997 The classification of tight contact structures on the 3-torus Commun. Anal Geom. 5 413-38
43. Kuperberg G 1996 A volume-preserving counterexample to the Seifert conjecture Comment. Math. Helv. 71 70-97
44. Kuperberg G and Kuperberg K 1994 Generalized counterexamples to the Seifert conjecture Ann. Math. 144 239-68
45. Kuperberg K 1994 A smooth counterexample to the Seifert conjecture Ann. Math. 140 723-32
46. Libermann P and Marle Ch-M 1987 Symplectic Geometry and Analytic Mechanics (Mathematics and its Applications vol 35) (Boston, MA: Reidel)
47. Loss M and Yau H-T 1986 Stability of Coulomb systems with magnetic fields III: zero-energy bound states of the Pauli operator Commun. Math. Phys. 104 283-90
48. Lutz R 1977 Structures de contact sur les fibrés principaux en cercles de dimension trois Ann. Inst. Fourier 3 1-15
49. Marsh G 1996 Force-free Magnetic Fields: Solutions, Topology and Applications (Singapore: World Scientific)
50. Martinet J 1971 Forms de Contact sur les Variétés de Dimension 3 (Springer Lecture Notes in Mathematics vol 209) (Berlin: Springer) pp 142-63
51. McDuff D and Salamon D 1995 Introduction to Symplectic Topology (New York: Oxford University Press)
52. McLaughlin D and Pirroneau O 1991 Some notes on periodic Beltrami fields in Cartesian geometry J. Math. Phys. 32 797-804
53. Moffatt H 1985 Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology: part I J. Fluid Mech. 159 359-78
54. Moser J 1965 On the volume elements on manifolds Trans. Am. Math. Soc. 120 280-96
55. Novikov S 1967 Topology of foliations Trans. Moscow Math. Soc. 14 268-304
56. Rabinowitz P 1979 Periodic solutions of Hamiltonian systems on a prescribed energy surface J. Differ. Equ. 33 336-52
57. Reeb G 1952 Sur certaines propriétés topologiques des trajectoires des systèmes dynamiques Acad. R. Belgique Cl. Sci. Mém. Coll. 27 1-64
58. Schweitzer P 1974 Counterexamples to the seifert conjecture and opening closed leaves of foliations Ann. Math. 100 386-400
59. Scott P 1983 The geometries of 3-manifolds Bull. London Math. Soc. 15 401-87
60. 2n Ann. Inst. H. Poincaré 4 337-356
61. Weinstein A 1979 On the hypothesis of Rabinowitz's periodic orbit theorem J. Differ. Equ. 33 353-8
62. George Zaslavsky M 1993 How long is the way from chaos to turbulence? New Approaches and Concepts in Turbulence (Monte Verità, 1991) (Basel: Birkhäuser) pp 165-89