Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Area comparison results for isotropic surfaces   Consider a 2-plane $P \subset \mathbb{C}^n$ and let $D$ be a bounded region
in $P$ with a piecewise-smooth boundary. Let $I(D)$ be the infimum of areas of
all piecewise-smooth isotropic surfaces in $\mathbb{C}^n$ with the same
boundary as $D$. Then $I(D)= \lambda_P^n \cdot Area(D)$. If $P$ is not complex,
$\lambda_P^n < \frac{3\pi}{2\sqrt{2}}$. For a complex plane $\mathbb{C} \subset
\mathbb{C}^n$, $\lambda_{\mathbb{C}}^n \geq 2$, $\lambda_{\mathbb{C}}^2 \geq 3$
and also $\frac{3\pi^2}{2\sqrt{2}}$ is the area of an explicit Hamiltonian
stationary isotropic Mobius band embedded in $\mathbb{C}^n$ whose boundary is a
unit circle in $\mathbb{C}$. As a corollary, a compact surface $\Sigma$
(possibly with boundary) in a symplectic manifold can be approximated by
isotropic surfaces of area $\leq \frac{3\pi}{2\sqrt{2}} Area(\Sigma)$. Another
corollary is that a closed curve of length $l$ in $\mathbb{C}^n$ bounds an
isotropic surface of area $\leq \frac{3l^2}{8\sqrt{2}}$. A related result is
the following: consider $\mathbb{C}P^1 \subset \mathbb{C}P^n$ and let $D$ be a
region in $\mathbb{C}P^1$. Let $I(D)$ be the infimum of areas of all isotropic
surfaces in $\mathbb{C}P^n$ with the same boundary as $D$ representing the same
relative homology class mod 2 as $D$. Then $ 2 \cdot Area(D) \leq I(D) \leq
\lambda_{\mathbb{C}}^n \cdot Area(D)$. Moreover the first inequality becomes an
equality for $D=\mathbb{C}P^1$.

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The categories of this paper are: nucl-ex
5
The categories of this paper are: gr-qc
22
The categories of this paper are: chao-dyn
12
The categories of this paper are: nucl-th
27
The categories of this paper are: comp-gas
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci
1
The categories of this paper are: cs
29
The categories of this paper are: funct-an
35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
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The categories of this paper are: eess
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The categories of this paper are: atom-ph