    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  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:
  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  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  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  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  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  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  M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on shape space of surfaces,, Journal of Geometric Mechanics, 3 (2011), 389. Google Scholar  M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on the manifold of all Riemannian metrics,, To appear in, ().   Google Scholar  M. Bauer, "Almost Local Metrics on Shape Space of Surfaces,", Ph.D thesis, (2010).   Google Scholar  A. L. Besse, "Einstein Manifolds,", Classics in Mathematics. Springer-Verlag, (2008). Google Scholar  P. Harms, "Sobolev Metrics on Shape Space of Surfaces,", Ph.D Thesis, (2010).   Google Scholar  P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms,, Doc. Math., 10 (2005), 217. Google Scholar  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  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  J. Peetre, Une caractérisation abstraite des opérateurs différentiels,, Math. Scand., 7 (1959), 211. Google Scholar  J. Peetre, Réctification à l'article "Une caractérisation abstraite des opérateurs différentiels",, Math. Scand., 8 (1960), 116. Google Scholar  J. Shah, $H^0$-type Riemannian metrics on the space of planar curves,, Quart. Appl. Math., 66 (2008), 123. Google Scholar  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  Jan Slovák, Peetre theorem for nonlinear operators,, Ann. Global Anal. Geom., 6 (1988), 273.  doi: 10.1007/BF00054575.  Google Scholar  A. Yezzi and A. Mennucci, Conformal riemannian metrics in space of curves,, EUSIPCO, (2004).   Google Scholar  A. Yezzi and A. Mennucci, Metrics in the space of curves,, \arXiv{math/0412454}, (2004).   Google Scholar  A. Yezzi and A. Mennucci, Conformal metrics and true "gradient flows" for curves,, in, 1 (2005), 913.   Google Scholar  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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