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]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361
[2]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382
[3]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302
[4]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248
[5]

Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377
[6]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142
[7]

Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161
[8]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049
[9]

Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 745-767. doi: 10.3934/dcdss.2020365
[10]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[11]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037
[12]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355
[13]

Azmy S. Ackleh, Nicolas Saintier. Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1469-1497. doi: 10.3934/dcdsb.2020169
[14]

Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021006
[15]

Zheng Han, Daoyuan Fang. Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020287
[16]

Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021022
[17]

Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325
[18]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020390
[19]

Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052
[20]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

2019 Impact Factor: 0.649

Tools

Metrics

  • PDF downloads (66)
  • HTML views (0)
  • Cited by (44)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences