Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Listening to the shape of a drum   The aim of this work is to link the quasiconformal geometry of a Euclidean
domain $U$ to the spectral properties of its Dirichlet integral $\D$, through
the algebra of multipliers $\M(H^{1,2}(U))$ of the Sobolev space. In the main
result we prove that a homeomorphism $\gamma:U\to\gamma(U)$ between Euclidean
domains, giving rise to an algebraic isomorphism $a\mapsto a\circ\gamma$
between $\M(H^{1,2}(\gamma(V)))$ and $\M(H^{1,2}(V))$ for any relatively
compact domain $V\subseteq U$ and leaving invariant the corresponding
fundamental tones (first non zero eigenvalues) of $\D$ \[
\mu_1(\gamma(V),a)=\mu_1(V,a\circ\gamma)\, , \] is quasiconformal. A companion
characterization hold true for bounded distortion maps. In the converse
direction we prove that for $n\ge 3$\\ i) the M\"obius group $G(\R^n)$ acts
isometrically on the algebra of multipliers $\M(H^{1,2}_e(\R^n))$ of the
extended space $H^{1,2}_e(\R^n)$\\ ii) $(\D,H^{1,2}(\R^n))$ is a closable
quadratic form on $L^2(\R^n,\Gamma[a])$ with respect to the energy measure
$\Gamma[a]=|\nabla a|^2\, dx$ of any $a\in \M(H^{1,2}_e(\R^n))$\\ iii) for any
$\gamma\in G(\R^n)$, the form closure $(\D,\F^a)$ of $(\D,H^{1,2}(\R^n))$ is a
Dirichlet form on $L^2(\R^n,\Gamma[a])$, unitarily equivalent to
$(\D,\F^{a\circ\gamma})$ on $L^2(\R^n,\Gamma[a\circ\gamma])$. The results are
based on connections between fundamental tones and ergodic properties of
multipliers: in particular, it is shown that the fundamental tone of
$(\D,\F^a)$ on $L^2(U,\Gamma[a])$ is non vanishing $\mu_1(U,a)>0$ for any fully
supported $a\in\M(H^{1,2}(U))$, provided there exists a spectral gap for the
usual Laplacian.

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The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
23
The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc
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The categories of this paper are: funct-an
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The categories of this paper are: ao-sci
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The categories of this paper are: eess
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The categories of this paper are: adap-org
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The categories of this paper are: nlin
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The categories of this paper are: chem-ph
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The categories of this paper are: econ
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-lat
3
The categories of this paper are: physics
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The categories of this paper are: alg-geom