# American Institute of Mathematical Sciences

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

## 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

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.
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. [2] M. Bauer, P. Harms and P. W. Michor, Almost local metrics on shape space of hypersurfaces in n-space,, preprint, (2010). [3] Martin Bauer, "Almost Local Metrics on Shape Space of Surfaces,'', Ph.D thesis, (2010). [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). [11] Shoshichi Kobayashi and Katsumi Nomizu, "Foundations of Differential Geometry," Vol. I,, Wiley Classics Library, (1996). [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). [22] Steven Verpoort, "The Geometry of the Second Fundamental Form: Curvature Properties and Variational Aspects,'', Ph.D thesis, (2008). [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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##### 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). [3] Martin Bauer, "Almost Local Metrics on Shape Space of Surfaces,'', Ph.D thesis, (2010). [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). [11] Shoshichi Kobayashi and Katsumi Nomizu, "Foundations of Differential Geometry," Vol. I,, Wiley Classics Library, (1996). [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). [22] Steven Verpoort, "The Geometry of the Second Fundamental Form: Curvature Properties and Variational Aspects,'', Ph.D thesis, (2008). [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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