Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Index Characterization for Free Boundary Minimal Surfaces   In this paper, we compute the Morse index for a free boundary minimal
submanifold from data of two simpler problems. The first one is the
corresponding problem with fixed boundary condition; and the second is
associated with the Dirichlet-to-Neumann map for Jacobi fields. As an
application, we show that the Morse index of a free boundary minimal annulus is
equal to 4 if and only if it is the critical catenoid.

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The categories of this paper are: q-bio
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The categories of this paper are: hep-lat
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The categories of this paper are: astro-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg
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The categories of this paper are: adap-org
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: math-ph
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The categories of this paper are: q-fin