Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Dirichlet problem for curve shortening flow   We investigate the evolution of open curves with fixed endpoints under the
curve shortening flow, which evolves curves in proportion to their curvature.
Using a distance comparison of Huisken, we determine the long-term behavior of
open curves with fixed endpoints evolving in certain convex domains on surfaces
of constant curvature. Specifically, we show that such curves do not develop
singularities, and evolve to geodesics.

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The categories of this paper are: cs
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The categories of this paper are: adap-org
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: physics
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The categories of this paper are: comp-gas
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The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg
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The categories of this paper are: nlin
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The categories of this paper are: funct-an
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The categories of this paper are: ao-sci
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The categories of this paper are: stat
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The categories of this paper are: q-bio
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The categories of this paper are: dg-ga
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The categories of this paper are: supr-con
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The categories of this paper are: q-fin