American Institute of Mathematical Sciences

Advanced
December  2011, 3(4): 389-438. doi: 10.3934/jgm.2011.3.389

Sobolev metrics on shape space of surfaces

Martin Bauer 1, , Philipp Harms 2, and  Peter W. Michor 1,

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

Received  September 2010 Revised  August 2011 Published  February 2012

Let $M$ and $N$ be connected manifolds without boundary with $\dim(M) < \dim(N)$, and let $M$ compact. Then shape space in this work is either the manifold of submanifolds of $N$ that are diffeomorphic to $M$, or the orbifold of unparametrized immersions of $M$ in $N$. We investigate the Sobolev Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions: $$ G^P_f(h,k) = \int_{M} \overline{g}( P^fh, k) vol (f^*\overline{g})$$ where $\overline{g}$ is some fixed metric on $N$, $f^*\overline{g}$ is the induced metric on $M$, $h,k \in \Gamma(f^*TN)$ are tangent vectors at $f$ to the space of embeddings or immersions, and $P^f$ is a positive, selfadjoint, bijective scalar pseudo differential operator of order $2p$ depending smoothly on $f$. We consider later specifically the operator $P^f=1 + A\Delta^p$, where $\Delta$ is the Bochner-Laplacian on $M$ induced by the metric $f^*\overline{g}$. For these metrics we compute the geodesic equations both on the space of immersions and on shape space, and also the conserved momenta arising from the obvious symmetries. We also show that the geodesic equation is well-posed on spaces of immersions, and also on diffeomorphism groups. We give examples of numerical solutions.
Keywords: shape space, well-posedness, Sobolev type metric, geodesic equation., Surface matching.
Mathematics Subject Classification: Primary: 58B20, 58D15, 58E1.
Citation: Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space of surfaces. Journal of Geometric Mechanics, 2011, 3 (4) : 389-438. doi: 10.3934/jgm.2011.3.389
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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

Jürgen Eichhorn, "Global Analysis on Open Manifolds,'', Nova Science Publishers, (2007).   Google Scholar

[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.  Google Scholar

[9]

François Gay-Balmaz, Well-posedness of higher dimensional Camassa-Holm equations,, Bull. Transilv. Univ. Braşov Ser. III, 2(51) (2009), 55.   Google Scholar

[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).   Google Scholar

[13]

Andreas Kriegl and Peter W. Michor, "The Convenient Setting of Global Analysis,'', Mathematical Surveys and Monographs, 53 (1997).   Google Scholar

[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.  Google Scholar

[15]

Peter W. Michor, Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach,, in, 69 (2006), 133.   Google Scholar

[16]

Peter W. Michor, "Topics in Differential Geometry,'', Graduate Studies in Mathematics, 93 (2008).   Google Scholar

[17]

Peter W. Michor and David Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms,, Doc. Math., 10 (2005), 217.   Google Scholar

[18]

Peter W. Michor and David Mumford, Riemannian geometries on spaces of plane curves,, J. Eur. Math. Soc. (JEMS), 8 (2006), 1.   Google Scholar

[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.  Google Scholar

[20]

M. A. Shubin, "Pseudodifferential Operators and Spectral Theory,'', Springer Series in Soviet Mathematics, (1987).   Google Scholar

[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.   Google Scholar

[24]

Laurent Younes, Computable elastic distances between shapes,, SIAM J. Appl. Math., 58 (1998), 565.  doi: 10.1137/S0036139995287685.  Google Scholar

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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

Jürgen Eichhorn, "Global Analysis on Open Manifolds,'', Nova Science Publishers, (2007).   Google Scholar

[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.  Google Scholar

[9]

François Gay-Balmaz, Well-posedness of higher dimensional Camassa-Holm equations,, Bull. Transilv. Univ. Braşov Ser. III, 2(51) (2009), 55.   Google Scholar

[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).   Google Scholar

[13]

Andreas Kriegl and Peter W. Michor, "The Convenient Setting of Global Analysis,'', Mathematical Surveys and Monographs, 53 (1997).   Google Scholar

[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.  Google Scholar

[15]

Peter W. Michor, Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach,, in, 69 (2006), 133.   Google Scholar

[16]

Peter W. Michor, "Topics in Differential Geometry,'', Graduate Studies in Mathematics, 93 (2008).   Google Scholar

[17]

Peter W. Michor and David Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms,, Doc. Math., 10 (2005), 217.   Google Scholar

[18]

Peter W. Michor and David Mumford, Riemannian geometries on spaces of plane curves,, J. Eur. Math. Soc. (JEMS), 8 (2006), 1.   Google Scholar

[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.  Google Scholar

[20]

M. A. Shubin, "Pseudodifferential Operators and Spectral Theory,'', Springer Series in Soviet Mathematics, (1987).   Google Scholar

[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.   Google Scholar

[24]

Laurent Younes, Computable elastic distances between shapes,, SIAM J. Appl. Math., 58 (1998), 565.  doi: 10.1137/S0036139995287685.  Google Scholar

[1]

Ming Wang. Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5763-5788. doi: 10.3934/dcds.2016053
[2]

Alexander V. Rezounenko, Petr Zagalak. Non-local PDEs with discrete state-dependent delays: Well-posedness in a metric space. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 819-835. doi: 10.3934/dcds.2013.33.819
[3]

Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673
[4]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic & Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032
[5]

G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327
[6]

Luiz Gustavo Farah. Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1521-1539. doi: 10.3934/cpaa.2009.8.1521
[7]

Seckin Demirbas. Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1517-1530. doi: 10.3934/cpaa.2017072
[8]

P. Blue, J. Colliander. Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS. Communications on Pure & Applied Analysis, 2006, 5 (4) : 691-708. doi: 10.3934/cpaa.2006.5.691
[9]

Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095
[10]

Hartmut Pecher. Corrigendum of "Local well-posedness for the nonlinear Dirac equation in two space dimensions". Communications on Pure & Applied Analysis, 2015, 14 (2) : 737-742. doi: 10.3934/cpaa.2015.14.737
[11]

Zhaohi Huo, Yueling Jia, Qiaoxin Li. Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1797-1851. doi: 10.3934/dcdss.2016075
[12]

Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007
[13]

Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241
[14]

Nils Strunk. Well-posedness for the supercritical gKdV equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 527-542. doi: 10.3934/cpaa.2014.13.527
[15]

A. Alexandrou Himonas, Curtis Holliman. On well-posedness of the Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 469-488. doi: 10.3934/dcds.2011.31.469
[16]

Haydi Israel. Well-posedness and long time behavior of an Allen-Cahn type equation. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2811-2827. doi: 10.3934/cpaa.2013.12.2811
[17]

Jun-ichi Segata. Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1093-1105. doi: 10.3934/dcds.2010.27.1093
[18]

Yoshihiro Shibata. Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface. Evolution Equations & Control Theory, 2018, 7 (1) : 117-152. doi: 10.3934/eect.2018007
[19]

Daniel Coutand, Steve Shkoller. A simple proof of well-posedness for the free-surface incompressible Euler equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 429-449. doi: 10.3934/dcdss.2010.3.429
[20]

Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315

2018 Impact Factor: 0.525

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences