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

  • MARTINS BRUVERIS (a1), PETER W. MICHOR (a2) and DAVID MUMFORD (a3)
Abstract

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.

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The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
References
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