  In 1961 G.Polya published a paper about the eigenvalues of vibrating
membrane. The "free vibrating membrane"' corresponds to the Neumann-Laplace
operator in bounded plane domains. In this paper we obtain estimates for the
first eigenvalue of this operator in a large class of domains that we call as
conformal regular domains, that includes convex domains, John domains etc... On
the base 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 that is based on weighted
Poincar\'e-Sobolev inequalities obtained by the authors recently.
