Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Knots, minimal surfaces and J-holomorphic curves   Let $K$ be a link in the 3-sphere, viewed as the ideal boundary of hyperbolic
4-space $\mathbb{H}^4$. Our main result is that the number of minimal surfaces
in $\mathbb{H}^4$ with ideal boundary $K$ is a link invariant. I.e. the number
of minimal surfaces (with fixed genus) is finite and doesn't change under
isotopies of $K$. To define the count, the link must be generic in a certain
sense and then minimal surfaces are counted with an appropriate sign. These
counts actually give a family of link invariants, indexed by the genus of the
filling and a second integer describing the extrinsic topology of how the
surface sits in $\mathbb{H}^4$. These invariants can be seen as Gromov-Witten
invariants counting J-holomorphic curves in the twistor space $Z$ of
$\mathbb{H}^4$. Whilst Gromov-Witten theory suggests the general scheme for
constructing the minimal surface link-invariants, there are substantial
differences in how this must be carried out in our situation. These are due to
the fact that the geometry of both $\mathbb{H}^4$ and $Z$ becomes singular at
infinity, and so the J-holomorphic curve equation is degenerate, rather than
elliptic, at the boundary. This means that both the Fredholm and compactness
arguments involve completely new features, in some places more complicated and
in others simpler, when compared with the usual story.

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The categories of this paper are: mtrl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: math-ph
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The categories of this paper are: eess
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The categories of this paper are: hep-lat
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The categories of this paper are: cs
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The categories of this paper are: gr-qc
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The categories of this paper are: alg-geom
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin
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The categories of this paper are: cmp-lg
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The categories of this paper are: plasm-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
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The categories of this paper are: acc-phys