  We study spectral stability estimates of the Dirichlet eigenvalues of the
Laplacian in non-convex domains $\Omega\subset\mathbb R^2$. With the help of
these estimates we obtain asymptotically sharp inequalities of ratios of
eigenvalues in the frameworks of the Payne-P\'olya-Weinberger inequalities.
These estimates are equivalent to spectral gap estimates of the Dirichlet
eigenvalues of the Laplacian in non-convex domains in terms of conformal
(hyperbolic) geometry.
