Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Free-Boundary Problems for Holomorphic Curves in the 6-Sphere   We remark on two different free-boundary problems for holomorphic curves in
nearly-K\"{a}hler 6-manifolds. First, we observe that a holomorphic curve in a
geodesic ball $B$ of the round 6-sphere that meets $\partial B$ orthogonally
must be totally geodesic. Consequently, we obtain rigidity results for
reflection-invariant holomorphic curves in $\mathbb{S}^6$ and associative cones
in $\mathbb{R}^7$.
  Second, we consider holomorphic curves with boundary on a Lagrangian
submanifold in a strict nearly-K\"{a}hler 6-manifold. By deriving a suitable
second variation formula for area, we observe a topological lower bound on the
Morse index. In both settings, our methods are complex-geometric, closely
following arguments of Fraser-Schoen and Chen-Fraser.

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The categories of this paper are: hep-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: stat
22
The categories of this paper are: chao-dyn
17
The categories of this paper are: q-fin
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The categories of this paper are: patt-sol
12
The categories of this paper are: nucl-th
28
The categories of this paper are: alg-geom
31
The categories of this paper are: ao-sci
36
The categories of this paper are: cmp-lg
35
The categories of this paper are: mtrl-th
37
The categories of this paper are: supr-con
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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: econ
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The categories of this paper are: math-ph