Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-22T16:37:30.487Z Has data issue: false hasContentIssue false
  • English
  • Français

GEODESIC COMPLETENESS FOR SOBOLEV METRICS ON THE SPACE OF IMMERSED PLANE CURVES

Part of: Spaces and manifolds of mappings Classical differential geometry General higher-order equations and systems Infinite-dimensional manifolds

Published online by Cambridge University Press:  28 July 2014

MARTINS BRUVERIS Open the ORCID record for MARTINS BRUVERIS [Opens in a new window] ,
PETER W. MICHOR  and
DAVID MUMFORD
MARTINS BRUVERIS
Affiliation:
Institut de Mathématiques, EPFL, Lausanne 1015, Switzerland; [email protected]
PETER W. MICHOR
Affiliation:
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, Wien 1090, Austria
DAVID MUMFORD
Affiliation:
Division of Applied Mathematics, Brown University, Box F, Providence, RI 02912, USA

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as for initial data in certain Sobolev spaces. Thus the space of closed plane curves equipped with such a metric is geodesically complete. We find lower bounds for the geodesic distance in terms of curvature and its derivatives.

MSC classification

Secondary: 58D15: Manifolds of mappings 35G55: Initial value problems for nonlinear higher-order systems 53A04: Curves in Euclidean space 58B20: Riemannian, Finsler and other geometric structures
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e19
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author(s) 2014

References

Adams, R. A., Sobolev Spaces, 2nd edn (Academic Press, 2003).Google Scholar
Bauer, M., Bruveris, M. and Michor, P. W., ‘Overview of the geometries of shape spaces and diffeomorphism groups’, J. Math. Imaging Vision (2014), doi: 10.1007/s10851-013-0490-z.Google Scholar
Bauer, M., Bruveris, M. and Michor, P. W., ‘R-transforms for Sobolev H 2-metrics on spaces of plane curves’, Geom. Imaging Comput. 1 (1) (2014), 1–56.Google Scholar
Bauer, M., Harms, P. and Michor, P. W., ‘Sobolev metrics on shape space of surfaces’, J. Geom. Mech. 3 (4) (2011), 389–438.Google Scholar
Bauer, M., Harms, P. and Michor, P. W., ‘Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics’, J. Geom. Mech. 4 (4) (2012), 365–383.Google Scholar
Bruveris, M. and Vialard, F.-X., ‘On completeness of groups of diffeomorphisms’, (2014), preprint: arXiv:1403.2089.Google Scholar
Charpiat, G., Maurel, P., Pons, J.-P., Keriven, R. and Faugeras, O., ‘Generalized gradients: Priors on minimization flows’, Int. J. Comput. Vis. 73 (3) (2007), 325–344.Google Scholar
Dieudonné, J., Foundations of Modern Analysis, Enlarged and corrected printing, Pure and Applied Mathematics, vol. 10-I (Academic Press, New York, 1969.Google Scholar
Ebin, D. G. and Marsden, J., ‘Groups of diffeomorphisms and the motion of an incompressible fluid’, Ann. of Math. (2) 92 (1970), 102–163.Google Scholar
Faà di Bruno, F., ‘Note sur une nouvelle formule du calcul difféntielle’, Quart. J. Math. 1 (1855), 359–360.Google Scholar
Inci, H., Kappeler, T. and Topalov, P., On the Regularity of the Composition of Diffeomorphisms, Memoirs of the American Mathematical Society, vol. 226 (American Mathematical Society, 2013).Google Scholar
Jones, G. S., ‘Fundamental inequalities for discrete and discontinuous functional equations’, J. Soc. Ind. Appl. Maths 12 (1964), 43–57.Google Scholar
Kriegl, A. and Michor, P. W., The convenient setting of global analysis, Mathematical Surveys and Monographs, vol. 53 (American Mathematical Society, Providence, RI, 1997).Google Scholar
Kurtek, S., Srivastava, A., Klassen, E. and Ding, Z., ‘Statistical modeling of curves using shapes and related features’, J. Am. Stat. Assoc. 107 (499) (2012), 1152–1165.Google Scholar
Mennucci, A., Yezzi, A. and Sundaramoorthi, G., ‘Properties of Sobolev-type metrics in the space of curves’, Interfaces Free Bound. 10 (4) (2008), 423–445.Google Scholar
Micheli, M., Michor, P. W. and Mumford, D., ‘Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds’, Izv. Math. 77 (3) (2013), 541–570.Google Scholar
Michor, P. W. and Mumford, D., ‘Riemannian geometries on spaces of plane curves’, J. Eur. Math. Soc. (JEMS) 8 (2006), 1–48.Google Scholar
Michor, P. W. and Mumford, D., ‘An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach’, Appl. Comput. Harmon. Anal. 23 (1) (2007), 74–113.Google Scholar
Mumford, D. and Michor, P. W., ‘On Euler’s equation and ’EPDiff’’, J. Geom. Mech. 5 (3) (2013), 319–344.Google Scholar
Nirenberg, L., ‘On elliptic partial differential equations’, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115–162.Google Scholar
Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (Eds.), NIST Handbook of Mathematical Functions (US Department of Commerce National Institute of Standards and Technology, Washington, DC, 2010).Google Scholar
Pachpatte, B. G., Inequalities for differential and integral equations, Mathematics in Science and Engineering, vol. 197 (Academic Press Inc., San Diego, CA, 1998).Google Scholar
Sundaramoorthi, G., Mennucci, A., Soatto, S. and Yezzi, A., ‘A new geometric metric in the space of curves, and applications to tracking deforming objects by prediction and filtering’, SIAM J. Imaging Sci. 4 (1) (2011), 109–145.Google Scholar
Sundaramoorthi, G., Yezzi, A. and Mennucci, A. C., ‘Sobolev active contours’, Int. J. Comput. Vis. 73 (3) (2007), 345–366.Google Scholar
Trouvé, A. and Younes, L., ‘Local geometry of deformable templates’, SIAM J. Math. Anal. 37 (1) (2005), 17–59. electronic.Google Scholar