Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Li-Yau inequality for the 1-dimensional Willmore energy   By the classical Li-Yau inequality, an immersion of a closed surface in
$\mathbb{R}^n$ with Willmore energy below $8\pi$ has to be embedded. We discuss
analogous results for curves in $\mathbb{R}^2$, involving Euler's elastic
energy and other possible curvature functionals. Additionally, we provide
applications to associated gradient flows.

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The categories of this paper are: supr-con
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The categories of this paper are: alg-geom
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The categories of this paper are: acc-phys
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The categories of this paper are: dg-ga
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The categories of this paper are: solv-int
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: econ
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The categories of this paper are: gr-qc
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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: math-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: ao-sci