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In 1961 G.Polya published a paper about the eigenvalues of vibratingmembrane. The "free vibrating membrane"' corresponds to the Neumann-Laplaceoperator in bounded plane domains. In this paper we obtain estimates for thefirst eigenvalue of this operator in a large class of domains that we call asconformal regular domains, that includes convex domains, John domains etc... Onthe base of our estimates we conjecture that the eigenvalues of theNeumann-Laplace operator depend on the hyperbolic metrics of plane domains. Wepropose a new method for the estimates that is based on weightedPoincar\'e-Sobolev inequalities obtained by the authors recently.

On the First Eigenvalues of Free Vibrating Membrane in Conformal Regular Domains