Green functions and the Dirichlet spectrum

This article has results of four types. We show that the first eigenvalue $\lambda_{1}(\Omega)$ of the weighted Laplacian of a bounded domain with smooth boundary can be obtained by S. Sato's iteration scheme of the Green operator, taking the limit $\lambda_{1}(\Omega)=\lim_{k\to \infty} \Vert G^{k}(f)\Vert_{L^2}/\Vert G^{k+1}(f)\Vert_{L^2}$ for any $f\in L^{2}(\Omega, \mu)$, $f > 0$. Then, we study the $L^{1}(\Omega, \mu)$-moment spectrum of $\Omega$ in terms of iterates of the Green operator $G$, extending the work of McDonald–Meyers to the weighted setting. As corollary, we obtain the first eigenvalue of a weighted bounded domain in terms of the $L^{1}(\Omega, \mu)$-moment spectrum, generalizing the work of Hurtado–Markvorsen–Palmer. Finally, we study the radial spectrum $\sigma^{\rm rad}(B_{h}(o,r))$ of rotationally invariant geodesic balls $B_{h}(o,r)$ of model manifolds. We prove an identity relating the radial eigenvalues of $\sigma^{\rm rad}(B_{h}(o,r))$ to an isoperimetric quotient, i.e., $\sum 1/\lambda_{i}^{\rm rad} = \int V(s)/S(s) ds$, $V(s)={\rm vol}(B_{h}(o,s))$ and $S(s)={\rm vol}(\partial B_{h}(o,s))$. We then consider a proper minimal surface $M\subset \mathbb{R}^{3}$ and the extrinsic ball $\Omega=M\cap B_{\mathbb{R}^{3}}(o,r)$. We obtain upper and lower estimates for the series $\sum \lambda_i^{-2}(\Omega)$ in terms of the volume ${\rm vol}(\Omega)$ and the radius $r$ of the extrinsic ball $\Omega$.