Abstract
We consider the first non-zero eigenvalue λ1 of the Laplacian on hyperbolic surfaces for which one disconnecting collar degenerates and prove that essentially agrees with the dual of the differential of the degenerating Fenchel-Nielsen length coordinate. As a consequence, we can improve previous results of Schoen, Wolpert and Yau and of Burger to obtain estimates with optimal error rates, and obtain new information on the leading order terms of the polyhomogeneous expansion of λ1 of Albin, Rochon and Sher.
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