Sharp eigenvalue estimates on degenerating surfaces
We consider the first non-zero eigenvalue λ1 of the Laplacian on hyperbolic surfaces for which one disconnecting collar degenerates and prove that 8π∇ log(λ1) essentially agrees with the dual of the differential of the degenerating Fenchel-Nielsen length coordinate. As a consequence, we can improve...
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| Format: | Journal article |
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Taylor and Francis
2019
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| Summary: | We consider the first non-zero eigenvalue λ1 of the Laplacian on hyperbolic surfaces for which one disconnecting collar degenerates and prove that 8π∇ log(λ1) 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, Yau [33] and Burger [6] 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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