Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An Allard-type boundary regularity theorem for $2d$ minimizing currents
  at smooth curves with arbitrary multiplicity   We consider integral area-minimizing $2$-dimensional currents $T$ in
$U\subset \mathbb R^{2+n}$ with $\partial T = Q[\![\Gamma]\!]$, where $Q\in
\mathbb N \setminus \{0\}$ and $\Gamma$ is sufficiently smooth. We prove that,
if $q\in \Gamma$ is a point where the density of $T$ is strictly below
$\frac{Q+1}{2}$, then the current is regular at $q$. The regularity is
understood in the following sense: there is a neighborhood of $q$ in which $T$
consists of a finite number of regular minimal submanifolds meeting
transversally at $\Gamma$ (and counted with the appropriate integer
multiplicity). In view of well-known examples, our result is optimal, and it is
the first nontrivial generalization of a classical theorem of Allard for $Q=1$.
As a corollary, if $\Omega\subset \mathbb R^{2+n}$ is a bounded uniformly
convex set and $\Gamma\subset \partial \Omega$ a smooth $1$-dimensional closed
submanifold, then any area-minimizing current $T$ with $\partial T = Q
[\![\Gamma]\!]$ is regular in a neighborhood of $\Gamma$.

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The categories of this paper are: quant-ph
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The categories of this paper are: physics
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-th
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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: supr-con
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The categories of this paper are: q-bio
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The categories of this paper are: nucl-ex
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The categories of this paper are: funct-an
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The categories of this paper are: solv-int
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The categories of this paper are: gr-qc
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The categories of this paper are: q-alg
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The categories of this paper are: ao-sci