American Institute of Mathematical Sciences

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June  2015, 7(2): 125-150. doi: 10.3934/jgm.2015.7.125

Completeness properties of Sobolev metrics on the space of curves

Martins Bruveris 1,

1. 

Department of Mathematics, Brunel Unversity London, Uxbridge UB8 3PH, United Kingdom

Received  July 2014 Revised  March 2015 Published  June 2015

We study completeness properties of Sobolev metrics on the space of immersed curves and on the shape space of unparametrized curves. We show that Sobolev metrics of order $n\geq 2$ are metrically complete on the space $\mathcal{I}^n(S^1,\mathbb{R}^d)$ of Sobolev immersions of the same regularity and that any two curves in the same connected component can be joined by a minimizing geodesic. These results then imply that the shape space of unparametrized curves has the structure of a complete length space.
Keywords: Immersed curves, Sobolev metrics, minimizing geodesics, shape space., completeness.
Mathematics Subject Classification: Primary: 58D10; Secondary: 58B20, 53A04, 35A0.
Citation: Martins Bruveris. Completeness properties of Sobolev metrics on the space of curves. Journal of Geometric Mechanics, 2015, 7 (2) : 125-150. doi: 10.3934/jgm.2015.7.125
References:
[1]

R. A. Adams, Sobolev Spaces,, 2nd edition, (2003).   Google Scholar

[2]

H. Amann, Compact embeddings of vector-valued Sobolev and Besov spaces,, Dedicated to the memory of Branko Najman, 35(55) (2000), 161.   Google Scholar

[3]

M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics,, J. Geom. Mech., 4 (2012), 365.   Google Scholar

[4]

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 Anal. Geom., 41 (2012), 461.  doi: 10.1007/s10455-011-9294-9.  Google Scholar

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M. Bauer, M. Bruveris, S. Marsland and P. W. Michor, Constructing reparameterization invariant metrics on spaces of plane curves,, Differential Geom. Appl., 34 (2014), 139.  doi: 10.1016/j.difgeo.2014.04.008.  Google Scholar

[6]

M. Bauer, M. Bruveris and P. Michor, $R$-transforms for Sobolev $H^2$-metrics on spaces of plane curves,, Geom. Imaging Comput., 1 (2014), 1.  doi: 10.4310/GIC.2014.v1.n1.a1.  Google Scholar

[7]

M. Bauer, M. Bruveris and P. Michor, Overview of the geometries of shape spaces and diffeomorphism groups,, Journal of Mathematical Imaging and Vision, 50 (2014), 60.  doi: 10.1007/s10851-013-0490-z.  Google Scholar

[8]

M. Bauer and P. Harms, Metrics on spaces of immersions where horizontality equals normality,, Differential Geom. Appl., 39 (2015), 166.  doi: 10.1016/j.difgeo.2014.12.008.  Google Scholar

[9]

M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on shape space of surfaces,, J. Geom. Mech., 3 (2011), 389.   Google Scholar

[10]

M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on the manifold of all Riemannian metrics,, J. Differential Geom., 94 (2013), 187.   Google Scholar

[11]

M. Bruveris, P. W. Michor and D. Mumford, Geodesic completeness for Sobolev metrics on the space of immersed plane curves,, Forum Math. Sigma, 2 (2014).  doi: 10.1017/fms.2014.19.  Google Scholar

[12]

M. Bruveris and F.-X. Vialard, On completeness of groups of diffeomorphisms,, preprint, (2014).   Google Scholar

[13]

D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry,, Graduate Studies in Mathematics, (2001).  doi: 10.1090/gsm/033.  Google Scholar

[14]

A. Burtscher, Length structures on manifolds with continuous Riemannian metrics,, preprint, (2013).   Google Scholar

[15]

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.  Google Scholar

[16]

G. Charpiat, P. Maurel, J.-P. Pons, R. Keriven and O. Faugeras, Generalized gradients: Priors on minimization flows,, Int. J. Comput. Vision, 73 (2007), 325.  doi: 10.1007/s11263-006-9966-2.  Google Scholar

[17]

D. G. Ebin, The manifold of Riemannian metrics,, in Global Analysis (Proc. Sympos. Pure Math., (1968), 11.   Google Scholar

[18]

D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. of Math. (2), 92 (1970), 102.  doi: 10.2307/1970699.  Google Scholar

[19]

J. Eells, Jr., A setting for global analysis,, Bull. Amer. Math. Soc., 72 (1966), 751.  doi: 10.1090/S0002-9904-1966-11558-6.  Google Scholar

[20]

H. I. Elíasson, Condition (C) and geodesics on Sobolev manifolds,, Bull. Amer. Math. Soc., 77 (1971), 1002.  doi: 10.1090/S0002-9904-1971-12836-7.  Google Scholar

[21]

F. Gay-Balmaz, Well-posedness of higher dimensional Camassa-Holm equations,, Bull. Transilv. Univ. Braşov Ser. III, 2 (2009), 55.   Google Scholar

[22]

R. S. Hamilton, The inverse function theorem of Nash and Moser,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65.  doi: 10.1090/S0273-0979-1982-15004-2.  Google Scholar

[23]

D. Huet, Décomposition Spectrale et Opérateurs,, Le Mathématicien, (1976).   Google Scholar

[24]

H. Inci, T. Kappeler and P. Topalov, On the regularity of the composition of diffeomorphisms,, Mem. Amer. Math. Soc., 226 (2013).  doi: 10.1090/S0065-9266-2013-00676-4.  Google Scholar

[25]

G. S. Jones, Fundamental inequalities for discrete and discontinuous functional equations,, J. Soc. Indust. Appl. Math., 12 (1964), 43.  doi: 10.1137/0112004.  Google Scholar

[26]

T. Kappeler, E. Loubet and P. Topalov, Riemannian exponential maps of the diffeomorphism groups of $\mathbbT^2$,, Asian J. Math., 12 (2008), 391.  doi: 10.4310/AJM.2008.v12.n3.a7.  Google Scholar

[27]

W. P. A. Klingenberg, Riemannian Geometry,, 2nd edition, (1995).  doi: 10.1515/9783110905120.  Google Scholar

[28]

H. Kodama and P. W. Michor, The homotopy type of the space of degree 0-immersed plane curves,, Rev. Mat. Complut., 19 (2006), 227.  doi: 10.5209/rev_REMA.2006.v19.n1.16660.  Google Scholar

[29]

S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group,, J. Math. Phys., 40 (1999), 857.  doi: 10.1063/1.532690.  Google Scholar

[30]

A. Kriegl and P. W. Michor, The Convenient Setting of Global Analysis,, Mathematical Surveys and Monographs, (1997).  doi: 10.1090/surv/053.  Google Scholar

[31]

S. Lang, Fundamentals of Differential Geometry,, Graduate Texts in Mathematics, (1999).  doi: 10.1007/978-1-4612-0541-8.  Google Scholar

[32]

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.  Google Scholar

[33]

P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms,, Doc. Math., 10 (2005), 217.   Google Scholar

[34]

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

[35]

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

[36]

G. Nardi, G. Peyré and F.-X. Vialard, Geodesics on Shape Spaces with Bounded Variation and Sobolev Metrics,, Technical report, (2014).   Google Scholar

[37]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (eds.), NIST Handbook of Mathematical Functions,, U.S. Department of Commerce National Institute of Standards and Technology, (2010).   Google Scholar

[38]

B. G. Pachpatte, Inequalities for Differential and Integral Equations,, Mathematics in Science and Engineering, (1998).   Google Scholar

[39]

R. S. Palais, Foundations of Global Non-Linear Analysis,, W. A. Benjamin, (1968).   Google Scholar

[40]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis,, 2nd edition, (1980).   Google Scholar

[41]

M. Rumpf and B. Wirth, Variational time discretization of geodesic calculus,, IMA J. Numer. Anal., (2014).  doi: 10.1093/imanum/dru027.  Google Scholar

[42]

J. Shah, $H^0$-type Riemannian metrics on the space of planar curves,, Quart. Appl. Math., 66 (2008), 123.  doi: 10.1090/S0033-569X-07-01084-4.  Google Scholar

[43]

J. Shah, An $H^2$ Riemannian metric on the space of planar curves modulo similitudes,, Adv. in Appl. Math., 51 (2013), 483.  doi: 10.1016/j.aam.2013.06.003.  Google Scholar

[44]

N. K. Smolentsev, Diffeomorphism groups of compact manifolds,, Sovrem. Mat. Prilozh., (2006), 3.  doi: 10.1007/s10958-007-0471-0.  Google Scholar

[45]

A. Srivastava, E. Klassen, S. H. Joshi and I. H. Jermyn, Shape analysis of elastic curves in Euclidean spaces,, IEEE T. Pattern Anal., 33 (2011), 1415.  doi: 10.1109/TPAMI.2010.184.  Google Scholar

[46]

G. Sundaramoorthi, A. Yezzi and A. C. Mennucci, Sobolev active contours,, in Variational, (3752), 109.  doi: 10.1007/11567646_10.  Google Scholar

[47]

K. Yosida, Functional Analysis,, Sixth edition, (1980).   Google Scholar

[48]

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.  doi: 10.4171/RLM/506.  Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces,, 2nd edition, (2003).   Google Scholar

[2]

H. Amann, Compact embeddings of vector-valued Sobolev and Besov spaces,, Dedicated to the memory of Branko Najman, 35(55) (2000), 161.   Google Scholar

[3]

M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics,, J. Geom. Mech., 4 (2012), 365.   Google Scholar

[4]

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 Anal. Geom., 41 (2012), 461.  doi: 10.1007/s10455-011-9294-9.  Google Scholar

[5]

M. Bauer, M. Bruveris, S. Marsland and P. W. Michor, Constructing reparameterization invariant metrics on spaces of plane curves,, Differential Geom. Appl., 34 (2014), 139.  doi: 10.1016/j.difgeo.2014.04.008.  Google Scholar

[6]

M. Bauer, M. Bruveris and P. Michor, $R$-transforms for Sobolev $H^2$-metrics on spaces of plane curves,, Geom. Imaging Comput., 1 (2014), 1.  doi: 10.4310/GIC.2014.v1.n1.a1.  Google Scholar

[7]

M. Bauer, M. Bruveris and P. Michor, Overview of the geometries of shape spaces and diffeomorphism groups,, Journal of Mathematical Imaging and Vision, 50 (2014), 60.  doi: 10.1007/s10851-013-0490-z.  Google Scholar

[8]

M. Bauer and P. Harms, Metrics on spaces of immersions where horizontality equals normality,, Differential Geom. Appl., 39 (2015), 166.  doi: 10.1016/j.difgeo.2014.12.008.  Google Scholar

[9]

M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on shape space of surfaces,, J. Geom. Mech., 3 (2011), 389.   Google Scholar

[10]

M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on the manifold of all Riemannian metrics,, J. Differential Geom., 94 (2013), 187.   Google Scholar

[11]

M. Bruveris, P. W. Michor and D. Mumford, Geodesic completeness for Sobolev metrics on the space of immersed plane curves,, Forum Math. Sigma, 2 (2014).  doi: 10.1017/fms.2014.19.  Google Scholar

[12]

M. Bruveris and F.-X. Vialard, On completeness of groups of diffeomorphisms,, preprint, (2014).   Google Scholar

[13]

D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry,, Graduate Studies in Mathematics, (2001).  doi: 10.1090/gsm/033.  Google Scholar

[14]

A. Burtscher, Length structures on manifolds with continuous Riemannian metrics,, preprint, (2013).   Google Scholar

[15]

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.  Google Scholar

[16]

G. Charpiat, P. Maurel, J.-P. Pons, R. Keriven and O. Faugeras, Generalized gradients: Priors on minimization flows,, Int. J. Comput. Vision, 73 (2007), 325.  doi: 10.1007/s11263-006-9966-2.  Google Scholar

[17]

D. G. Ebin, The manifold of Riemannian metrics,, in Global Analysis (Proc. Sympos. Pure Math., (1968), 11.   Google Scholar

[18]

D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. of Math. (2), 92 (1970), 102.  doi: 10.2307/1970699.  Google Scholar

[19]

J. Eells, Jr., A setting for global analysis,, Bull. Amer. Math. Soc., 72 (1966), 751.  doi: 10.1090/S0002-9904-1966-11558-6.  Google Scholar

[20]

H. I. Elíasson, Condition (C) and geodesics on Sobolev manifolds,, Bull. Amer. Math. Soc., 77 (1971), 1002.  doi: 10.1090/S0002-9904-1971-12836-7.  Google Scholar

[21]

F. Gay-Balmaz, Well-posedness of higher dimensional Camassa-Holm equations,, Bull. Transilv. Univ. Braşov Ser. III, 2 (2009), 55.   Google Scholar

[22]

R. S. Hamilton, The inverse function theorem of Nash and Moser,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65.  doi: 10.1090/S0273-0979-1982-15004-2.  Google Scholar

[23]

D. Huet, Décomposition Spectrale et Opérateurs,, Le Mathématicien, (1976).   Google Scholar

[24]

H. Inci, T. Kappeler and P. Topalov, On the regularity of the composition of diffeomorphisms,, Mem. Amer. Math. Soc., 226 (2013).  doi: 10.1090/S0065-9266-2013-00676-4.  Google Scholar

[25]

G. S. Jones, Fundamental inequalities for discrete and discontinuous functional equations,, J. Soc. Indust. Appl. Math., 12 (1964), 43.  doi: 10.1137/0112004.  Google Scholar

[26]

T. Kappeler, E. Loubet and P. Topalov, Riemannian exponential maps of the diffeomorphism groups of $\mathbbT^2$,, Asian J. Math., 12 (2008), 391.  doi: 10.4310/AJM.2008.v12.n3.a7.  Google Scholar

[27]

W. P. A. Klingenberg, Riemannian Geometry,, 2nd edition, (1995).  doi: 10.1515/9783110905120.  Google Scholar

[28]

H. Kodama and P. W. Michor, The homotopy type of the space of degree 0-immersed plane curves,, Rev. Mat. Complut., 19 (2006), 227.  doi: 10.5209/rev_REMA.2006.v19.n1.16660.  Google Scholar

[29]

S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group,, J. Math. Phys., 40 (1999), 857.  doi: 10.1063/1.532690.  Google Scholar

[30]

A. Kriegl and P. W. Michor, The Convenient Setting of Global Analysis,, Mathematical Surveys and Monographs, (1997).  doi: 10.1090/surv/053.  Google Scholar

[31]

S. Lang, Fundamentals of Differential Geometry,, Graduate Texts in Mathematics, (1999).  doi: 10.1007/978-1-4612-0541-8.  Google Scholar

[32]

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.  Google Scholar

[33]

P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms,, Doc. Math., 10 (2005), 217.   Google Scholar

[34]

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

[35]

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

[36]

G. Nardi, G. Peyré and F.-X. Vialard, Geodesics on Shape Spaces with Bounded Variation and Sobolev Metrics,, Technical report, (2014).   Google Scholar

[37]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (eds.), NIST Handbook of Mathematical Functions,, U.S. Department of Commerce National Institute of Standards and Technology, (2010).   Google Scholar

[38]

B. G. Pachpatte, Inequalities for Differential and Integral Equations,, Mathematics in Science and Engineering, (1998).   Google Scholar

[39]

R. S. Palais, Foundations of Global Non-Linear Analysis,, W. A. Benjamin, (1968).   Google Scholar

[40]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis,, 2nd edition, (1980).   Google Scholar

[41]

M. Rumpf and B. Wirth, Variational time discretization of geodesic calculus,, IMA J. Numer. Anal., (2014).  doi: 10.1093/imanum/dru027.  Google Scholar

[42]

J. Shah, $H^0$-type Riemannian metrics on the space of planar curves,, Quart. Appl. Math., 66 (2008), 123.  doi: 10.1090/S0033-569X-07-01084-4.  Google Scholar

[43]

J. Shah, An $H^2$ Riemannian metric on the space of planar curves modulo similitudes,, Adv. in Appl. Math., 51 (2013), 483.  doi: 10.1016/j.aam.2013.06.003.  Google Scholar

[44]

N. K. Smolentsev, Diffeomorphism groups of compact manifolds,, Sovrem. Mat. Prilozh., (2006), 3.  doi: 10.1007/s10958-007-0471-0.  Google Scholar

[45]

A. Srivastava, E. Klassen, S. H. Joshi and I. H. Jermyn, Shape analysis of elastic curves in Euclidean spaces,, IEEE T. Pattern Anal., 33 (2011), 1415.  doi: 10.1109/TPAMI.2010.184.  Google Scholar

[46]

G. Sundaramoorthi, A. Yezzi and A. C. Mennucci, Sobolev active contours,, in Variational, (3752), 109.  doi: 10.1007/11567646_10.  Google Scholar

[47]

K. Yosida, Functional Analysis,, Sixth edition, (1980).   Google Scholar

[48]

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.  doi: 10.4171/RLM/506.  Google Scholar

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