Listening to the shape of a drum
Fabio E.G. Cipriani, J.-L. Sauvageot

TL;DR
This work links the quasiconformal geometry of Euclidean domains to spectral properties of the Dirichlet integral via multiplier algebras, characterizing quasiconformal maps through algebraic isomorphisms that preserve fundamental tones.
Contribution
It provides a novel characterization of quasiconformal maps using algebraic isomorphisms of multiplier algebras and spectral invariants, extending to bounded distortion maps and higher dimensions.
Findings
Quasiconformal maps correspond to algebraic isomorphisms preserving spectral properties.
The M"obius group acts isometrically on multiplier algebras in erent spaces.
Dirichlet forms associated with multipliers are unitarily equivalent under M"obius transformations.
Abstract
The aim of this work is to link the quasiconformal geometry of a Euclidean domain to the spectral properties of its Dirichlet integral , through the algebra of multipliers of the Sobolev space. In the main result we prove that a homeomorphism between Euclidean domains, giving rise to an algebraic isomorphism between and for any relatively compact domain and leaving invariant the corresponding fundamental tones (first non zero eigenvalues) of \[ \mu_1(\gamma(V),a)=\mu_1(V,a\circ\gamma)\, , \] is quasiconformal. A companion characterization hold true for bounded distortion maps. In the converse direction we prove that for \\ i) the M\"obius group acts isometrically on the algebra of multipliers of the extended space…
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Taxonomy
TopicsAnalytic and geometric function theory · Spectral Theory in Mathematical Physics · Nonlinear Partial Differential Equations
