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Fredholm properties of the $L^{2}$ exponential map on the symplectomorphism group
1. | 3 Hardie Street, Palmerston North, Hokowhitu, 4410, New Zealand |
References:
[1] |
V. I. Arnold, On the Differential Geometry of Infinite-Dimensional Lie Groups and its appliction to the Hydrodynamics of Perfect Fluids,, Vladimir I. Arnold: Collected Works, (2014). Google Scholar |
[2] |
V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics,, Springer-Verlag, (1998).
|
[3] |
D. Bao, J. Lafontaine and T. Ratiu, On a non-linear equation related to the geometry of the diffeomorphism group,, Pacific Journal of Mathematics, 158 (1993), 223.
doi: 10.2140/pjm.1993.158.223. |
[4] |
D. Ebin, Geodesics on the symplectomorphism group,, Journal of Geometric and Functional Analysis, 22 (2012), 202.
doi: 10.1007/s00039-012-0150-2. |
[5] |
D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Annals of Mathematics, 92 (1970), 102.
doi: 10.2307/1970699. |
[6] |
D. Ebin, G. Misiołek and S. Preston, Singularities of the exponential map on the volume-preserving diffeomorphism group,, Journal of Geometric and Functional Analysis, 16 (2006), 850.
doi: 10.1007/s00039-006-0573-8. |
[7] |
D. Holm and C. Tronci, The geodesic vlasov equation and its integrable moment closures,, Journal of Geometric Mechanics, 1 (2009), 181.
doi: 10.3934/jgm.2009.1.181. |
[8] |
H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics,, Springer, (1994).
doi: 10.1007/978-3-0348-8540-9. |
[9] |
T. Kato, Perturbation Theory for Linear Operators,, Springer-Verlag, (1966).
|
[10] |
J. Marsden and A. Weinstein, The Hamiltonian structure of the Maxwell-Vlasov equations,, Physica D, 4 (1982), 394.
doi: 10.1016/0167-2789(82)90043-4. |
[11] |
J. Marsden, A. Weinstein, R. Schmid and R. Spencer, Hamiltonian systems and symmetry groups with applications to plasma physics,, Atti della Accademia delle Scienze di Torino. Classe di Scienze Fisiche, 117 (1983), 289.
|
[12] |
G. Misiolek, Stability of flows of ideal fluids and the geometry of the groups of diffeomorphisms,, Indiana University Mathematics Journal, 42 (1993), 215.
doi: 10.1512/iumj.1993.42.42011. |
[13] |
G. Misiołek, Conjugate points in $\mathcalD_{\mu}(\mathbbT^{2})$,, Proceedings of the American Mathematics Society, 124 (1996), 977.
doi: 10.1090/S0002-9939-96-03149-8. |
[14] |
G. Misiołek and S. Preston, Fredholm properties of riemannian exponential maps on diffeomorphism groups,, Inventiones Mathematicae, 179 (2010), 191.
doi: 10.1007/s00222-009-0217-3. |
[15] |
C. B. Morrey, Multiple Integrals in the Calculus of Variations,, Springer-Verlag, (1966).
|
[16] |
P. J. Morrison, The Maxwell-Vlasov equations as a continuous Hamiltonian system,, Physics Letters A, 80 (1980), 383.
doi: 10.1016/0375-9601(80)90776-8. |
[17] |
S. Preston, For ideal fluids, Eulerian and Lagrangian instabilities are equivalent,, Journal of Geometric and Functional Analysis, 14 (2004), 1044.
doi: 10.1007/s00039-004-0482-7. |
[18] |
S. Preston, On the volumorphism group, the first conjugate point is the hardest,, Communications in Mathematical Physics, 267 (2006), 493.
doi: 10.1007/s00220-006-0070-9. |
[19] |
T. Ratiu and R. Schmid, Three remarkable diffeomorphism groups,, Mathematische Zeitschrift, 177 (1981), 81.
doi: 10.1007/BF01214340. |
[20] |
R. Schmid, Infinite dimensional lie groups and algebras in mathematical physics,, Hindawi Advances in Mathematical Physics, 2010 (2010).
|
[21] |
A. Shnirelman, Generalized fluid flows, their approximations and applications,, Journal of Geometric and Functional Analysis, 4 (1994), 586.
doi: 10.1007/BF01896409. |
[22] |
S. Smale, An infinite dimensional version of Sard's Theorem,, American Journal of Mathematics, 87 (1965), 861.
doi: 10.2307/2373250. |
[23] |
N. K. Smolentsev, A Biinvariant Metric on the Group of Symplectic Diffeomorphisms and the Equation $\partial_t\DeltaF = {\DeltaF,F}$,, Translated from Sibirskii Matematicheskii Shurnal, 27 (1986), 150.
|
[24] |
M. Taylor, Partial Differential Equations I, Basic Theory,, Springer, (2011).
doi: 10.1007/978-1-4419-7055-8. |
[25] |
I. Ustilovsky, Conjugate points on geodesics of hofer's metric,, Differential Geometry and its Applications, 6 (1996), 327.
doi: 10.1016/S0926-2245(96)00027-7. |
show all references
References:
[1] |
V. I. Arnold, On the Differential Geometry of Infinite-Dimensional Lie Groups and its appliction to the Hydrodynamics of Perfect Fluids,, Vladimir I. Arnold: Collected Works, (2014). Google Scholar |
[2] |
V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics,, Springer-Verlag, (1998).
|
[3] |
D. Bao, J. Lafontaine and T. Ratiu, On a non-linear equation related to the geometry of the diffeomorphism group,, Pacific Journal of Mathematics, 158 (1993), 223.
doi: 10.2140/pjm.1993.158.223. |
[4] |
D. Ebin, Geodesics on the symplectomorphism group,, Journal of Geometric and Functional Analysis, 22 (2012), 202.
doi: 10.1007/s00039-012-0150-2. |
[5] |
D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Annals of Mathematics, 92 (1970), 102.
doi: 10.2307/1970699. |
[6] |
D. Ebin, G. Misiołek and S. Preston, Singularities of the exponential map on the volume-preserving diffeomorphism group,, Journal of Geometric and Functional Analysis, 16 (2006), 850.
doi: 10.1007/s00039-006-0573-8. |
[7] |
D. Holm and C. Tronci, The geodesic vlasov equation and its integrable moment closures,, Journal of Geometric Mechanics, 1 (2009), 181.
doi: 10.3934/jgm.2009.1.181. |
[8] |
H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics,, Springer, (1994).
doi: 10.1007/978-3-0348-8540-9. |
[9] |
T. Kato, Perturbation Theory for Linear Operators,, Springer-Verlag, (1966).
|
[10] |
J. Marsden and A. Weinstein, The Hamiltonian structure of the Maxwell-Vlasov equations,, Physica D, 4 (1982), 394.
doi: 10.1016/0167-2789(82)90043-4. |
[11] |
J. Marsden, A. Weinstein, R. Schmid and R. Spencer, Hamiltonian systems and symmetry groups with applications to plasma physics,, Atti della Accademia delle Scienze di Torino. Classe di Scienze Fisiche, 117 (1983), 289.
|
[12] |
G. Misiolek, Stability of flows of ideal fluids and the geometry of the groups of diffeomorphisms,, Indiana University Mathematics Journal, 42 (1993), 215.
doi: 10.1512/iumj.1993.42.42011. |
[13] |
G. Misiołek, Conjugate points in $\mathcalD_{\mu}(\mathbbT^{2})$,, Proceedings of the American Mathematics Society, 124 (1996), 977.
doi: 10.1090/S0002-9939-96-03149-8. |
[14] |
G. Misiołek and S. Preston, Fredholm properties of riemannian exponential maps on diffeomorphism groups,, Inventiones Mathematicae, 179 (2010), 191.
doi: 10.1007/s00222-009-0217-3. |
[15] |
C. B. Morrey, Multiple Integrals in the Calculus of Variations,, Springer-Verlag, (1966).
|
[16] |
P. J. Morrison, The Maxwell-Vlasov equations as a continuous Hamiltonian system,, Physics Letters A, 80 (1980), 383.
doi: 10.1016/0375-9601(80)90776-8. |
[17] |
S. Preston, For ideal fluids, Eulerian and Lagrangian instabilities are equivalent,, Journal of Geometric and Functional Analysis, 14 (2004), 1044.
doi: 10.1007/s00039-004-0482-7. |
[18] |
S. Preston, On the volumorphism group, the first conjugate point is the hardest,, Communications in Mathematical Physics, 267 (2006), 493.
doi: 10.1007/s00220-006-0070-9. |
[19] |
T. Ratiu and R. Schmid, Three remarkable diffeomorphism groups,, Mathematische Zeitschrift, 177 (1981), 81.
doi: 10.1007/BF01214340. |
[20] |
R. Schmid, Infinite dimensional lie groups and algebras in mathematical physics,, Hindawi Advances in Mathematical Physics, 2010 (2010).
|
[21] |
A. Shnirelman, Generalized fluid flows, their approximations and applications,, Journal of Geometric and Functional Analysis, 4 (1994), 586.
doi: 10.1007/BF01896409. |
[22] |
S. Smale, An infinite dimensional version of Sard's Theorem,, American Journal of Mathematics, 87 (1965), 861.
doi: 10.2307/2373250. |
[23] |
N. K. Smolentsev, A Biinvariant Metric on the Group of Symplectic Diffeomorphisms and the Equation $\partial_t\DeltaF = {\DeltaF,F}$,, Translated from Sibirskii Matematicheskii Shurnal, 27 (1986), 150.
|
[24] |
M. Taylor, Partial Differential Equations I, Basic Theory,, Springer, (2011).
doi: 10.1007/978-1-4419-7055-8. |
[25] |
I. Ustilovsky, Conjugate points on geodesics of hofer's metric,, Differential Geometry and its Applications, 6 (1996), 327.
doi: 10.1016/S0926-2245(96)00027-7. |
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