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

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

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; martins.bruveris@epfl.ch
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

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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

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