Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Geodesic Completeness for Sobolev Metrics on the Space of Immersed Plane
  Curves   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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The categories of this paper are: adap-org
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The categories of this paper are: plasm-ph
3
The categories of this paper are: physics
28
The categories of this paper are: alg-geom
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The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-th
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The categories of this paper are: quant-ph
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The categories of this paper are: stat
1
The categories of this paper are: cs
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The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: econ
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The categories of this paper are: gr-qc
4
The categories of this paper are: cond-mat