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:
 [1] M. Bauer and M. Bruveris, A new Riemannian setting for surface registration, 3nd MICCAI Workshop on Mathematical Foundations of Computational Anatomy, (2011), 182-194. 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), 461-472. 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), 244-310. 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), 33-41. 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), 389-438.  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, University of Vienna, 2010. Google Scholar [10] A. L. Besse, "Einstein Manifolds," Classics in Mathematics. Springer-Verlag, Berlin, 2008.  Google Scholar [11] P. Harms, "Sobolev Metrics on Shape Space of Surfaces," Ph.D Thesis, University of Vienna, 2010. Google Scholar [12] P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. Math., 10 (2005), 217-245 (electronic).  Google Scholar [13] P. W. Michor and D. Mumford, Riemannian geometries on spaces of plane curves, J. Eur. Math. Soc. (JEMS), 8 (2006), 1-48. 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), 74-113. doi: 10.1016/j.acha.2006.07.004.  Google Scholar [15] J. Peetre, Une caractérisation abstraite des opérateurs différentiels, Math. Scand., 7 (1959), 211-218.  Google Scholar [16] J. Peetre, Réctification à l'article "Une caractérisation abstraite des opérateurs différentiels", Math. Scand., 8 (1960), 116-120.  Google Scholar [17] J. Shah, $H^0$-type Riemannian metrics on the space of planar curves, Quart. Appl. Math., 66 (2008), 123-137.  Google Scholar [18] M. A. Shubin, "Pseudodifferential Operators and Spectral Theory," Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 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), 273-283. 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, December (2004). Google Scholar [22] A. Yezzi and A. Mennucci, Conformal metrics and true "gradient flows" for curves, in "Proceedings of the Tenth IEEE International Conference on Computer Vision," 1, 913-919, Washington, (2005). IEEE Computer Society. 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), 25-57. doi: 10.4171/RLM/506.  Google Scholar

show all references

References:
 [1] M. Bauer and M. Bruveris, A new Riemannian setting for surface registration, 3nd MICCAI Workshop on Mathematical Foundations of Computational Anatomy, (2011), 182-194. 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), 461-472. 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), 244-310. 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), 33-41. 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), 389-438.  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, University of Vienna, 2010. Google Scholar [10] A. L. Besse, "Einstein Manifolds," Classics in Mathematics. Springer-Verlag, Berlin, 2008.  Google Scholar [11] P. Harms, "Sobolev Metrics on Shape Space of Surfaces," Ph.D Thesis, University of Vienna, 2010. Google Scholar [12] P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. Math., 10 (2005), 217-245 (electronic).  Google Scholar [13] P. W. Michor and D. Mumford, Riemannian geometries on spaces of plane curves, J. Eur. Math. Soc. (JEMS), 8 (2006), 1-48. 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), 74-113. doi: 10.1016/j.acha.2006.07.004.  Google Scholar [15] J. Peetre, Une caractérisation abstraite des opérateurs différentiels, Math. Scand., 7 (1959), 211-218.  Google Scholar [16] J. Peetre, Réctification à l'article "Une caractérisation abstraite des opérateurs différentiels", Math. Scand., 8 (1960), 116-120.  Google Scholar [17] J. Shah, $H^0$-type Riemannian metrics on the space of planar curves, Quart. Appl. Math., 66 (2008), 123-137.  Google Scholar [18] M. A. Shubin, "Pseudodifferential Operators and Spectral Theory," Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 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), 273-283. 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, December (2004). Google Scholar [22] A. Yezzi and A. Mennucci, Conformal metrics and true "gradient flows" for curves, in "Proceedings of the Tenth IEEE International Conference on Computer Vision," 1, 913-919, Washington, (2005). IEEE Computer Society. 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), 25-57. doi: 10.4171/RLM/506.  Google Scholar
 [1] Nobu Kishimoto, Minjie Shan, Yoshio Tsutsumi. Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1283-1307. doi: 10.3934/dcds.2020078 [2] Ming Wang. Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5763-5788. doi: 10.3934/dcds.2016053 [3] Alexander V. Rezounenko, Petr Zagalak. Non-local PDEs with discrete state-dependent delays: Well-posedness in a metric space. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 819-835. doi: 10.3934/dcds.2013.33.819 [4] Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673 [5] Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic & Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032 [6] G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327 [7] Luiz Gustavo Farah. Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1521-1539. doi: 10.3934/cpaa.2009.8.1521 [8] Seckin Demirbas. Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1517-1530. doi: 10.3934/cpaa.2017072 [9] Baoyan Sun, Kung-Chien Wu. Global well-posedness and exponential stability for the fermion equation in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021147 [10] Wei Yan, Yimin Zhang, Yongsheng Li, Jinqiao Duan. Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5825-5849. doi: 10.3934/dcds.2021097 [11] P. Blue, J. Colliander. Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS. Communications on Pure & Applied Analysis, 2006, 5 (4) : 691-708. doi: 10.3934/cpaa.2006.5.691 [12] Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095 [13] Hartmut Pecher. Corrigendum of "Local well-posedness for the nonlinear Dirac equation in two space dimensions". Communications on Pure & Applied Analysis, 2015, 14 (2) : 737-742. doi: 10.3934/cpaa.2015.14.737 [14] Zhaohi Huo, Yueling Jia, Qiaoxin Li. Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1797-1851. doi: 10.3934/dcdss.2016075 [15] Mohamad Darwich. Local and global well-posedness in the energy space for the dissipative Zakharov-Kuznetsov equation in 3D. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3715-3724. doi: 10.3934/dcdsb.2020087 [16] Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007 [17] Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036 [18] Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete & Continuous Dynamical Systems, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241 [19] A. Alexandrou Himonas, Curtis Holliman. On well-posedness of the Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 469-488. doi: 10.3934/dcds.2011.31.469 [20] Nils Strunk. Well-posedness for the supercritical gKdV equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 527-542. doi: 10.3934/cpaa.2014.13.527

2020 Impact Factor: 0.857