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

Completeness properties of Sobolev metrics on the space of curves

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.
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, Academic Press, 2003.

[2]

H. Amann, Compact embeddings of vector-valued Sobolev and Besov spaces, Dedicated to the memory of Branko Najman, Glas. Mat. Ser. III, 35(55) (2000), 161-177.

[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), 365-383.

[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), 461-472. doi: 10.1007/s10455-011-9294-9.

[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), 139-165. doi: 10.1016/j.difgeo.2014.04.008.

[6]

M. Bauer, M. Bruveris and P. Michor, $\mathbb{R}^{N}$-transforms for Sobolev $H^2$-metrics on spaces of plane curves, Geom. Imaging Comput., 1 (2014), 1-56. doi: 10.4310/GIC.2014.v1.n1.a1.

[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), 60-97. doi: 10.1007/s10851-013-0490-z.

[8]

M. Bauer and P. Harms, Metrics on spaces of immersions where horizontality equals normality, Differential Geom. Appl., 39 (2015), 166-183. doi: 10.1016/j.difgeo.2014.12.008.

[9]

M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on shape space of surfaces, J. Geom. Mech., 3 (2011), 389-438.

[10]

M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on the manifold of all Riemannian metrics, J. Differential Geom., 94 (2013), 187-208. Available from: http://projecteuclid.org/euclid.jdg/1367438647.

[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), e19 (38 pages). doi: 10.1017/fms.2014.19.

[12]

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

[13]

D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/033.

[14]

A. Burtscher, Length structures on manifolds with continuous Riemannian metrics, preprint, 2013.

[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), 391-401. doi: 10.1016/0926-2245(91)90015-2.

[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), 325-344. doi: 10.1007/s11263-006-9966-2.

[17]

D. G. Ebin, The manifold of Riemannian metrics, in Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, 11-40.

[18]

D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. (2), 92 (1970), 102-163. doi: 10.2307/1970699.

[19]

J. Eells, Jr., A setting for global analysis, Bull. Amer. Math. Soc., 72 (1966), 751-807. doi: 10.1090/S0002-9904-1966-11558-6.

[20]

H. I. Elíasson, Condition (C) and geodesics on Sobolev manifolds, Bull. Amer. Math. Soc., 77 (1971), 1002-1005. doi: 10.1090/S0002-9904-1971-12836-7.

[21]

F. Gay-Balmaz, Well-posedness of higher dimensional Camassa-Holm equations, Bull. Transilv. Univ. Braşov Ser. III, 2 (2009), 55-58.

[22]

R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65-222. doi: 10.1090/S0273-0979-1982-15004-2.

[23]

D. Huet, Décomposition Spectrale et Opérateurs, Le Mathématicien, No. 16. Presses Universitaires de France, Paris, 1976.

[24]

H. Inci, T. Kappeler and P. Topalov, On the regularity of the composition of diffeomorphisms, Mem. Amer. Math. Soc., 226 (2013), vi+60pp. doi: 10.1090/S0065-9266-2013-00676-4.

[25]

G. S. Jones, Fundamental inequalities for discrete and discontinuous functional equations, J. Soc. Indust. Appl. Math., 12 (1964), 43-57. doi: 10.1137/0112004.

[26]

T. Kappeler, E. Loubet and P. Topalov, Riemannian exponential maps of the diffeomorphism groups of $\mathbbT^2$, Asian J. Math., 12 (2008), 391-420. doi: 10.4310/AJM.2008.v12.n3.a7.

[27]

W. P. A. Klingenberg, Riemannian Geometry, 2nd edition, de Gruyter Studies in Mathematics, 1, Walter de Gruyter & Co., Berlin, 1995. doi: 10.1515/9783110905120.

[28]

H. Kodama and P. W. Michor, The homotopy type of the space of degree 0-immersed plane curves, Rev. Mat. Complut., 19 (2006), 227-234. doi: 10.5209/rev_REMA.2006.v19.n1.16660.

[29]

S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868. doi: 10.1063/1.532690.

[30]

A. Kriegl and P. W. Michor, The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, 53, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/053.

[31]

S. Lang, Fundamentals of Differential Geometry, Graduate Texts in Mathematics, 191, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0541-8.

[32]

A. Mennucci, A. Yezzi and G. Sundaramoorthi, Properties of Sobolev-type metrics in the space of curves, Interfaces Free Bound., 10 (2008), 423-445. doi: 10.4171/IFB/196.

[33]

P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. Math., 10 (2005), 217-245.

[34]

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.

[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), 74-113. doi: 10.1016/j.acha.2006.07.004.

[36]

G. Nardi, G. Peyré and F.-X. Vialard, Geodesics on Shape Spaces with Bounded Variation and Sobolev Metrics, Technical report, preprint hal-00952672, 2014.

[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, Washington, DC, 2010.

[38]

B. G. Pachpatte, Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering, 197, Academic Press Inc., San Diego, CA, 1998.

[39]

R. S. Palais, Foundations of Global Non-Linear Analysis, W. A. Benjamin, Inc., New York-Amsterdam, 1968.

[40]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, 2nd edition, Academic Press, Inc., New York, 1980.

[41]

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

[42]

J. Shah, $H^0$-type Riemannian metrics on the space of planar curves, Quart. Appl. Math., 66 (2008), 123-137. doi: 10.1090/S0033-569X-07-01084-4.

[43]

J. Shah, An $H^2$ Riemannian metric on the space of planar curves modulo similitudes, Adv. in Appl. Math., 51 (2013), 483-506. doi: 10.1016/j.aam.2013.06.003.

[44]

N. K. Smolentsev, Diffeomorphism groups of compact manifolds, Sovrem. Mat. Prilozh., (2006), 3-100. doi: 10.1007/s10958-007-0471-0.

[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), 1415-1428. doi: 10.1109/TPAMI.2010.184.

[46]

G. Sundaramoorthi, A. Yezzi and A. C. Mennucci, Sobolev active contours, in Variational, Geometric, and Level Set Methods in Computer Vision, Lecture Notes in Computer Science, 3752, Springer-Verlag, Berlin-Heidelberg, 2005, 109-120. doi: 10.1007/11567646_10.

[47]

K. Yosida, Functional Analysis, Sixth edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 123, Springer-Verlag, Berlin, 1980.

[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), 25-57. doi: 10.4171/RLM/506.

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, 2nd edition, Academic Press, 2003.

[2]

H. Amann, Compact embeddings of vector-valued Sobolev and Besov spaces, Dedicated to the memory of Branko Najman, Glas. Mat. Ser. III, 35(55) (2000), 161-177.

[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), 365-383.

[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), 461-472. doi: 10.1007/s10455-011-9294-9.

[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), 139-165. doi: 10.1016/j.difgeo.2014.04.008.

[6]

M. Bauer, M. Bruveris and P. Michor, $\mathbb{R}^{N}$-transforms for Sobolev $H^2$-metrics on spaces of plane curves, Geom. Imaging Comput., 1 (2014), 1-56. doi: 10.4310/GIC.2014.v1.n1.a1.

[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), 60-97. doi: 10.1007/s10851-013-0490-z.

[8]

M. Bauer and P. Harms, Metrics on spaces of immersions where horizontality equals normality, Differential Geom. Appl., 39 (2015), 166-183. doi: 10.1016/j.difgeo.2014.12.008.

[9]

M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on shape space of surfaces, J. Geom. Mech., 3 (2011), 389-438.

[10]

M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on the manifold of all Riemannian metrics, J. Differential Geom., 94 (2013), 187-208. Available from: http://projecteuclid.org/euclid.jdg/1367438647.

[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), e19 (38 pages). doi: 10.1017/fms.2014.19.

[12]

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

[13]

D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/033.

[14]

A. Burtscher, Length structures on manifolds with continuous Riemannian metrics, preprint, 2013.

[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), 391-401. doi: 10.1016/0926-2245(91)90015-2.

[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), 325-344. doi: 10.1007/s11263-006-9966-2.

[17]

D. G. Ebin, The manifold of Riemannian metrics, in Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, 11-40.

[18]

D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. (2), 92 (1970), 102-163. doi: 10.2307/1970699.

[19]

J. Eells, Jr., A setting for global analysis, Bull. Amer. Math. Soc., 72 (1966), 751-807. doi: 10.1090/S0002-9904-1966-11558-6.

[20]

H. I. Elíasson, Condition (C) and geodesics on Sobolev manifolds, Bull. Amer. Math. Soc., 77 (1971), 1002-1005. doi: 10.1090/S0002-9904-1971-12836-7.

[21]

F. Gay-Balmaz, Well-posedness of higher dimensional Camassa-Holm equations, Bull. Transilv. Univ. Braşov Ser. III, 2 (2009), 55-58.

[22]

R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65-222. doi: 10.1090/S0273-0979-1982-15004-2.

[23]

D. Huet, Décomposition Spectrale et Opérateurs, Le Mathématicien, No. 16. Presses Universitaires de France, Paris, 1976.

[24]

H. Inci, T. Kappeler and P. Topalov, On the regularity of the composition of diffeomorphisms, Mem. Amer. Math. Soc., 226 (2013), vi+60pp. doi: 10.1090/S0065-9266-2013-00676-4.

[25]

G. S. Jones, Fundamental inequalities for discrete and discontinuous functional equations, J. Soc. Indust. Appl. Math., 12 (1964), 43-57. doi: 10.1137/0112004.

[26]

T. Kappeler, E. Loubet and P. Topalov, Riemannian exponential maps of the diffeomorphism groups of $\mathbbT^2$, Asian J. Math., 12 (2008), 391-420. doi: 10.4310/AJM.2008.v12.n3.a7.

[27]

W. P. A. Klingenberg, Riemannian Geometry, 2nd edition, de Gruyter Studies in Mathematics, 1, Walter de Gruyter & Co., Berlin, 1995. doi: 10.1515/9783110905120.

[28]

H. Kodama and P. W. Michor, The homotopy type of the space of degree 0-immersed plane curves, Rev. Mat. Complut., 19 (2006), 227-234. doi: 10.5209/rev_REMA.2006.v19.n1.16660.

[29]

S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868. doi: 10.1063/1.532690.

[30]

A. Kriegl and P. W. Michor, The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, 53, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/053.

[31]

S. Lang, Fundamentals of Differential Geometry, Graduate Texts in Mathematics, 191, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0541-8.

[32]

A. Mennucci, A. Yezzi and G. Sundaramoorthi, Properties of Sobolev-type metrics in the space of curves, Interfaces Free Bound., 10 (2008), 423-445. doi: 10.4171/IFB/196.

[33]

P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. Math., 10 (2005), 217-245.

[34]

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.

[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), 74-113. doi: 10.1016/j.acha.2006.07.004.

[36]

G. Nardi, G. Peyré and F.-X. Vialard, Geodesics on Shape Spaces with Bounded Variation and Sobolev Metrics, Technical report, preprint hal-00952672, 2014.

[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, Washington, DC, 2010.

[38]

B. G. Pachpatte, Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering, 197, Academic Press Inc., San Diego, CA, 1998.

[39]

R. S. Palais, Foundations of Global Non-Linear Analysis, W. A. Benjamin, Inc., New York-Amsterdam, 1968.

[40]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, 2nd edition, Academic Press, Inc., New York, 1980.

[41]

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

[42]

J. Shah, $H^0$-type Riemannian metrics on the space of planar curves, Quart. Appl. Math., 66 (2008), 123-137. doi: 10.1090/S0033-569X-07-01084-4.

[43]

J. Shah, An $H^2$ Riemannian metric on the space of planar curves modulo similitudes, Adv. in Appl. Math., 51 (2013), 483-506. doi: 10.1016/j.aam.2013.06.003.

[44]

N. K. Smolentsev, Diffeomorphism groups of compact manifolds, Sovrem. Mat. Prilozh., (2006), 3-100. doi: 10.1007/s10958-007-0471-0.

[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), 1415-1428. doi: 10.1109/TPAMI.2010.184.

[46]

G. Sundaramoorthi, A. Yezzi and A. C. Mennucci, Sobolev active contours, in Variational, Geometric, and Level Set Methods in Computer Vision, Lecture Notes in Computer Science, 3752, Springer-Verlag, Berlin-Heidelberg, 2005, 109-120. doi: 10.1007/11567646_10.

[47]

K. Yosida, Functional Analysis, Sixth edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 123, Springer-Verlag, Berlin, 1980.

[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), 25-57. doi: 10.4171/RLM/506.

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