# American Institute of Mathematical Sciences

December  2012, 4(4): 365-383. doi: 10.3934/jgm.2012.4.365

## Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics

 1 Fakultät f¨ur Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria 2 EdLabs, Harvard University, 44 Brattle Street, Cambridge, MA 02138

Received  September 2011 Revised  May 2012 Published  January 2013

In continuation of [7] we discuss metrics of the form $$G^P_f(h,k)=\int_M \sum_{i=0}^p\Phi_i\big(Vol(f)\big)\ \bar{g}\big((P_i)_fh,k\big) vol(f^*\bar{g})$$ on the space of immersions $Imm(M,N)$ and on shape space $B_i(M,N)=Imm(M,N)/{Diff}(M)$. Here $(N,\bar{g})$ is a complete Riemannian manifold, $M$ is a compact manifold, $f:M\to N$ is an immersion, $h$ and $k$ are tangent vectors to $f$ in the space of immersions, $f^*\bar{g}$ is the induced Riemannian metric on $M$, $vol(f^*\bar{g})$ is the induced volume density on $M$, $Vol(f)=\int_M vol(f^*\bar{g})$, $\Phi_i$ are positive real-valued functions, and $(P_i)_f$ are operators like some power of the Laplacian $\Delta^{f^*\bar{g}}$. We derive the geodesic equations for these metrics and show that they are sometimes well-posed with the geodesic exponential mapping a local diffeomorphism. The new aspect here are the weights $\Phi_i(Vol(f))$ which we use to construct scale invariant metrics and order 0 metrics with positive geodesic distance. We treat several concrete special cases in detail.
Citation: Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics. Journal of Geometric Mechanics, 2012, 4 (4) : 365-383. doi: 10.3934/jgm.2012.4.365
##### References:
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##### References:
 [1] M. Bauer and M. Bruveris, A new Riemannian setting for surface registration,, 3nd MICCAI Workshop on Mathematical Foundations of Computational Anatomy, (2011), 182.   Google Scholar [2] M. Bauer, M. Bruveris, C. Cotter, S. Marsland and P. W. Michor, Constructing reparametrization invariant metrics on spaces of plane curves,, \arXiv{1207.5965}., ().   Google Scholar [3] M. Bauer, M. Bruveris, P. Harms and P. W. Michor, Vanishing geodesic distance for the riemannian metric with geodesic equation the KdV-equation,, Ann. Global Analysis Geom., 41 (2012), 461.  doi: 10.1007/s10455-011-9294-9.  Google Scholar [4] M. Bauer, M. Bruveris, P. Harms and P. W. Michor, Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group,, Ann. Glob. Anal. Geom., ().  doi: doi:10.1007/s10455-012-9353-x.  Google Scholar [5] M. Bauer, P. Harms and P. W. Michor, Almost local metrics on shape space of hypersurfaces in n-space,, SIAM J. Imaging Sci., 5 (2012), 244.  doi: 10.1137/100807983.  Google Scholar [6] M. Bauer, P. Harms and P. W. Michor, Curvature weighted metrics on shape space of hypersurfaces in n-space,, Differential Geometry and its Applications, 30 (2012), 33.  doi: 10.1016/j.difgeo.2011.10.002.  Google Scholar [7] M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on shape space of surfaces,, Journal of Geometric Mechanics, 3 (2011), 389.   Google Scholar [8] M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on the manifold of all Riemannian metrics,, To appear in, ().   Google Scholar [9] M. Bauer, "Almost Local Metrics on Shape Space of Surfaces,", Ph.D thesis, (2010).   Google Scholar [10] A. L. Besse, "Einstein Manifolds,", Classics in Mathematics. Springer-Verlag, (2008).   Google Scholar [11] P. Harms, "Sobolev Metrics on Shape Space of Surfaces,", Ph.D Thesis, (2010).   Google Scholar [12] P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms,, Doc. Math., 10 (2005), 217.   Google Scholar [13] P. W. Michor and D. Mumford, Riemannian geometries on spaces of plane curves,, J. Eur. Math. Soc. (JEMS), 8 (2006), 1.  doi: 10.4171/JEMS/37.  Google Scholar [14] P. W. Michor and D. 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 [15] J. Peetre, Une caractérisation abstraite des opérateurs différentiels,, Math. Scand., 7 (1959), 211.   Google Scholar [16] J. Peetre, Réctification à l'article "Une caractérisation abstraite des opérateurs différentiels",, Math. Scand., 8 (1960), 116.   Google Scholar [17] J. Shah, $H^0$-type Riemannian metrics on the space of planar curves,, Quart. Appl. Math., 66 (2008), 123.   Google Scholar [18] M. A. Shubin, "Pseudodifferential Operators and Spectral Theory,", Springer Series in Soviet Mathematics. Springer-Verlag, (1987).  doi: 10.1007/978-3-642-96854-9.  Google Scholar [19] Jan Slovák, Peetre theorem for nonlinear operators,, Ann. Global Anal. Geom., 6 (1988), 273.  doi: 10.1007/BF00054575.  Google Scholar [20] A. Yezzi and A. Mennucci, Conformal riemannian metrics in space of curves,, EUSIPCO, (2004).   Google Scholar [21] A. Yezzi and A. Mennucci, Metrics in the space of curves,, \arXiv{math/0412454}, (2004).   Google Scholar [22] A. Yezzi and A. Mennucci, Conformal metrics and true "gradient flows" for curves,, in, 1 (2005), 913.   Google Scholar [23] L. Younes, P. W. Michor, J. Shah and D. Mumford, A metric on shape space with explicit geodesics,, Rend. Lincei Mat. Appl., 9 (2008), 25.  doi: 10.4171/RLM/506.  Google Scholar
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