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.