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Sobolev metrics on shape space of surfaces
1. | Fakultät f¨ur Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria, Austria |
2. | EdLabs, Harvard University, 44 Brattle Street, Cambridge, MA 02138, United States |
References:
[1] |
V. I. Arnold, Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier (Grenoble), 16 (1966), 319.
doi: 10.5802/aif.233. |
[2] |
M. Bauer, P. Harms and P. W. Michor, Almost local metrics on shape space of hypersurfaces in n-space,, preprint, (2010). Google Scholar |
[3] |
Martin Bauer, "Almost Local Metrics on Shape Space of Surfaces,'', Ph.D thesis, (2010). Google Scholar |
[4] |
Arthur L. Besse, "Einstein Manifolds,'', Reprint of the 1987 edition, (1987).
|
[5] |
V. Cervera, F. Mascaró and P. W. Michor, The action of the diffeomorphism group on the space of immersions,, Differential Geom. Appl., 1 (1991), 391.
doi: 10.1016/0926-2245(91)90015-2. |
[6] |
Adrian Constantin and Boris Kolev, Geodesic flow on the diffeomorphism group of the circle,, Comment. Math. Helv., 78 (2003), 787.
doi: 10.1007/s00014-003-0785-6. |
[7] |
Jürgen Eichhorn, "Global Analysis on Open Manifolds,'', Nova Science Publishers, (2007).
|
[8] |
Jürgen Eichhorn and Jan Fricke, The module structure theorem for Sobolev spaces on open manifolds,, Math. Nachr., 194 (1998), 35.
doi: 10.1002/mana.19981940105. |
[9] |
François Gay-Balmaz, Well-posedness of higher dimensional Camassa-Holm equations,, Bull. Transilv. Univ. Braşov Ser. III, 2(51) (2009), 55.
|
[10] |
Philipp Harms, "Sobolev Metrics on Shape Space of Surfaces,'', Ph.D thesis, (2010). Google Scholar |
[11] |
Shoshichi Kobayashi and Katsumi Nomizu, "Foundations of Differential Geometry," Vol. I,, Wiley Classics Library, (1996). Google Scholar |
[12] |
I. Kolář, P. W. Michor and J. Slovák, "Natural Operations in Differential Geometry,'', Springer-Verlag, (1993).
|
[13] |
Andreas Kriegl and Peter W. Michor, "The Convenient Setting of Global Analysis,'', Mathematical Surveys and Monographs, 53 (1997).
|
[14] |
A. Mennucci, A. Yezzi and G. Sundaramoorthi, Properties of Sobolev-type metrics in the space of curves,, Interfaces Free Bound., 10 (2008), 423.
doi: 10.4171/IFB/196. |
[15] |
Peter W. Michor, Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach,, in, 69 (2006), 133.
|
[16] |
Peter W. Michor, "Topics in Differential Geometry,'', Graduate Studies in Mathematics, 93 (2008).
|
[17] |
Peter W. Michor and David Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms,, Doc. Math., 10 (2005), 217.
|
[18] |
Peter W. Michor and David Mumford, Riemannian geometries on spaces of plane curves,, J. Eur. Math. Soc. (JEMS), 8 (2006), 1.
|
[19] |
Peter W. Michor and David Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach,, Appl. Comput. Harmon. Anal., 23 (2007), 74.
doi: 10.1016/j.acha.2006.07.004. |
[20] |
M. A. Shubin, "Pseudodifferential Operators and Spectral Theory,'', Springer Series in Soviet Mathematics, (1987).
|
[21] |
Alain Trouvé and Laurent Younes, "Diffeomorphic Matching Problems in One Dimension: Designing and Minimizing Matching Functionals,", Computer Vision, (1842). Google Scholar |
[22] |
Steven Verpoort, "The Geometry of the Second Fundamental Form: Curvature Properties and Variational Aspects,'', Ph.D thesis, (2008). Google Scholar |
[23] |
L. Younes, P. W. Michor, J. Shah and D. Mumford, A metric on shape space with explicit geodesics,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (2008), 25.
|
[24] |
Laurent Younes, Computable elastic distances between shapes,, SIAM J. Appl. Math., 58 (1998), 565.
doi: 10.1137/S0036139995287685. |
show all references
References:
[1] |
V. I. Arnold, Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier (Grenoble), 16 (1966), 319.
doi: 10.5802/aif.233. |
[2] |
M. Bauer, P. Harms and P. W. Michor, Almost local metrics on shape space of hypersurfaces in n-space,, preprint, (2010). Google Scholar |
[3] |
Martin Bauer, "Almost Local Metrics on Shape Space of Surfaces,'', Ph.D thesis, (2010). Google Scholar |
[4] |
Arthur L. Besse, "Einstein Manifolds,'', Reprint of the 1987 edition, (1987).
|
[5] |
V. Cervera, F. Mascaró and P. W. Michor, The action of the diffeomorphism group on the space of immersions,, Differential Geom. Appl., 1 (1991), 391.
doi: 10.1016/0926-2245(91)90015-2. |
[6] |
Adrian Constantin and Boris Kolev, Geodesic flow on the diffeomorphism group of the circle,, Comment. Math. Helv., 78 (2003), 787.
doi: 10.1007/s00014-003-0785-6. |
[7] |
Jürgen Eichhorn, "Global Analysis on Open Manifolds,'', Nova Science Publishers, (2007).
|
[8] |
Jürgen Eichhorn and Jan Fricke, The module structure theorem for Sobolev spaces on open manifolds,, Math. Nachr., 194 (1998), 35.
doi: 10.1002/mana.19981940105. |
[9] |
François Gay-Balmaz, Well-posedness of higher dimensional Camassa-Holm equations,, Bull. Transilv. Univ. Braşov Ser. III, 2(51) (2009), 55.
|
[10] |
Philipp Harms, "Sobolev Metrics on Shape Space of Surfaces,'', Ph.D thesis, (2010). Google Scholar |
[11] |
Shoshichi Kobayashi and Katsumi Nomizu, "Foundations of Differential Geometry," Vol. I,, Wiley Classics Library, (1996). Google Scholar |
[12] |
I. Kolář, P. W. Michor and J. Slovák, "Natural Operations in Differential Geometry,'', Springer-Verlag, (1993).
|
[13] |
Andreas Kriegl and Peter W. Michor, "The Convenient Setting of Global Analysis,'', Mathematical Surveys and Monographs, 53 (1997).
|
[14] |
A. Mennucci, A. Yezzi and G. Sundaramoorthi, Properties of Sobolev-type metrics in the space of curves,, Interfaces Free Bound., 10 (2008), 423.
doi: 10.4171/IFB/196. |
[15] |
Peter W. Michor, Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach,, in, 69 (2006), 133.
|
[16] |
Peter W. Michor, "Topics in Differential Geometry,'', Graduate Studies in Mathematics, 93 (2008).
|
[17] |
Peter W. Michor and David Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms,, Doc. Math., 10 (2005), 217.
|
[18] |
Peter W. Michor and David Mumford, Riemannian geometries on spaces of plane curves,, J. Eur. Math. Soc. (JEMS), 8 (2006), 1.
|
[19] |
Peter W. Michor and David Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach,, Appl. Comput. Harmon. Anal., 23 (2007), 74.
doi: 10.1016/j.acha.2006.07.004. |
[20] |
M. A. Shubin, "Pseudodifferential Operators and Spectral Theory,'', Springer Series in Soviet Mathematics, (1987).
|
[21] |
Alain Trouvé and Laurent Younes, "Diffeomorphic Matching Problems in One Dimension: Designing and Minimizing Matching Functionals,", Computer Vision, (1842). Google Scholar |
[22] |
Steven Verpoort, "The Geometry of the Second Fundamental Form: Curvature Properties and Variational Aspects,'', Ph.D thesis, (2008). Google Scholar |
[23] |
L. Younes, P. W. Michor, J. Shah and D. Mumford, A metric on shape space with explicit geodesics,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (2008), 25.
|
[24] |
Laurent Younes, Computable elastic distances between shapes,, SIAM J. Appl. Math., 58 (1998), 565.
doi: 10.1137/S0036139995287685. |
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