Closed Geodesics and Huber’s Theorem
There is no general procedure to be found in the literature which allows us to write down the spectrum of the Laplacian of a compact Riemann surface in closed form, say as a function of the moduli. Nevertheless, isospectrality and related problems can be solved by quite explicit geometric constructions. This is possible through Huber’s theorem, which states that the spectra of the Laplacian for two compact Riemann surfaces are the same if and only if the surfaces have the same length spectrum, i.e. the same sequence of lengths of closed geodesics.
KeywordsConjugacy Class Heat Kernel Fundamental Domain Trace Formula Counting Function
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