For hiperbolic plane $H$, let $$ \lambda_0(H)=\inf \frac{\int ||\nabla\phi||^2}{\int\phi^2} $$ where $\phi$ runs over all non-zero smooth functions on $H$ with compact support in the interior of $H$. How to show $\lambda_0(H)=\frac{1}{4}$ ?
2026-03-28 22:29:55.1774736995
How to show the bottom of $L^2$ spectrum of Laplacian on hyperbolic plane $H$ is $1/4$?
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