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
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show all references

References:
[1]

Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361. doi: 10.5802/aif.233.  Google Scholar

[2]

preprint, 2010, arXiv:math/1001.0717. Google Scholar

[3]

Ph.D thesis, University of Vienna, 2010. Google Scholar

[4]

Reprint of the 1987 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2008.  Google Scholar

[5]

Differential Geom. Appl., 1 (1991), 391-401. doi: 10.1016/0926-2245(91)90015-2.  Google Scholar

[6]

Comment. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6.  Google Scholar

[7]

Nova Science Publishers, Inc., New York, 2007.  Google Scholar

[8]

Math. Nachr., 194 (1998), 35-47. doi: 10.1002/mana.19981940105.  Google Scholar

[9]

Bull. Transilv. Univ. Braşov Ser. III, 2(51) (2009), 55-58.  Google Scholar

[10]

Ph.D thesis, University of Vienna, 2010. Google Scholar

[11]

Wiley Classics Library, John Wiley & Sons, Inc., New York, 1996. Google Scholar

[12]

Springer-Verlag, Berlin, 1993.  Google Scholar

[13]

Mathematical Surveys and Monographs, 53, American Mathematical Society, Providence, RI, 1997.  Google Scholar

[14]

Interfaces Free Bound., 10 (2008), 423-445. doi: 10.4171/IFB/196.  Google Scholar

[15]

in "Phase Space Analysis of Partial Differential Equations," Progr. Nonlinear Differential Equations Appl., 69, Birkhäuser Boston, Boston, MA, (2006), 133-215.  Google Scholar

[16]

Graduate Studies in Mathematics, 93, American Mathematical Society, Providence, RI, 2008.  Google Scholar

[17]

Doc. Math., 10 (2005), 217-245 (electronic).  Google Scholar

[18]

J. Eur. Math. Soc. (JEMS), 8 (2006), 1-48.  Google Scholar

[19]

Appl. Comput. Harmon. Anal., 23 (2007), 74-113. doi: 10.1016/j.acha.2006.07.004.  Google Scholar

[20]

Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987.  Google Scholar

[21]

Computer Vision, Vol. 1842, ECCV, 2000. Google Scholar

[22]

Ph.D thesis, Katholieke Universiteit Leuven, 2008. Google Scholar

[23]

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (2008), 25-57.  Google Scholar

[24]

SIAM J. Appl. Math., 58 (1998), 565-586 (electronic). doi: 10.1137/S0036139995287685.  Google Scholar

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