I am interested in examples of bounded open subsets of the hyperbolic space, for which the first eigenvalue of the Dirichlet Laplace operator (acting on functions) is known. In Euclidean space several such examples exist (e.g. rectangles, balls).

Probably there are also results for some domains in the hyperbolic space, but I could not find anything in the literature. Do you know some references, where I can find related calculations? For example, is the first eigenvalue for discs in hyperbolic plane calculated somewhere? Or is there any good approximation of the first eigenvalue of any domain?

I would appreciate your help very much.


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