Completeness properties of Sobolev metrics on the space of curves

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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

Abstract
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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, 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, $R$-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, $R$-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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