In 1961 G. Polya published a paper about the eigenvalues of vibrating membranes. The “free vibrating membrane” corresponds to the Neumann–Laplace operator in bounded plane domains. In this paper we obtain estimates for the firstnon-trivial eigenvalue of this operator in a large class of domains that we call conformal regular domains. This class includes convex domains, John domains etc. On the basis of our estimates we conjecture that the eigenvalues of the Neumann–Laplace operator depend on the hyperbolic metrics of plane domains. We propose a new method for the estimates which is based on weighted Poincaré–Sobolev inequalities, obtained by the authors recently.