On conformal spectral gap estimates of the Dirichlet-Laplacian

We study spectral stability estimates of the Dirichlet eigenvalues of the Laplacian in nonconvex domains $ \Omega \subset \mathbb{R}^2$. With the help of these estimates, we obtain asymptotically sharp inequalities of ratios of eigenvalues in the framework of the Payne-Pólya-Weinberger inequalities. These estimates are equivalent to spectral gap estimates of the Dirichlet eigenvalues of the Laplacian in nonconvex domains in terms of conformal (hyperbolic) geometry.