Geodesic Completeness for Sobolev Metrics on the Space of Immersed Plane Curves
Analysis of PDEs
2014-10-07 v2 Differential Geometry
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.
Keywords
Cite
@article{arxiv.1312.4995,
title = {Geodesic Completeness for Sobolev Metrics on the Space of Immersed Plane Curves},
author = {Martins Bruveris and Peter W. Michor and David Mumford},
journal= {arXiv preprint arXiv:1312.4995},
year = {2014}
}
Comments
36 pages, LaTeX