Listening to the shape of a drum

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 $U$ 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.