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

By Martins Bruveris, Peter W. Michor, David Mumford. Forum of Mathematics, Sigma, 2013

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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.

Subfields: Applied Mathematics, Computational Mechanics, Management, Monitoring, Policy and Law

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