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arXiv · 1312.4995

Geodesic Completeness for Sobolev Metrics on the Space of Immersed Plane Curves

Martins Bruveris·Peter W. Michor·David Mumford
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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 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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Martins Bruveris, Peter W. Michor, David Mumford. 2014-06-23. Geodesic Completeness for Sobolev Metrics on the Space of Immersed Plane Curves. https://doi.org/10.1017/fms.2014.19

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