Sharp eigenvalue estimates on degenerating surfaces

  title={Sharp eigenvalue estimates on degenerating surfaces},
  author={Nadine Grosse and Melanie Rupflin},
We consider the first non-zero eigenvalue $\lambda_1$ of the Laplacian on hyperbolic surfaces for which one disconnecting collar degenerates and prove that $8\pi \nabla\log(\lambda_1)$ essentially agrees with the dual of the differential of the degenerating Fenchel-Nielson length coordinate. As a corollary of our analysis, which is based in particular on the fine properties of holomorphic quadratic differentials from the joint works of Topping and the second author, we can improve previous… CONTINUE READING
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