On Conformal Spectral Gap Estimates of the Dirichlet-Laplacian

We study spectral stability estimates of the Dirichlet eigenvalues of the Laplacian in non-convex domains $\mathrm{\Omega }\subset {\mathbb{R}}^2$. 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.