# Sharp eigenvalue estimates on degenerating surfaces.

• Melanie Rupflin, University of Oxford
• Monday 28 January 2019, 15:00-16:00
• CMS, MR13.

We consider the first non-zero eigenvalue $\lambda_1$ of the Laplacian on hyperbolic surfaces for which one disconnecting collar degenerates and show that the gradient of $\lambda_1$ is given essentially explicitly in terms of the dual of the differential of the degenerating length coordinate. As a corollary we obtain sharp error estimates on $\lambda_1$ that improve previous results of Burger and Schoen-Wolpert-Yau and provide new information on the second term in the polyhomogenous expansion of $\lambda_1$. The presented results are joint work with Nadine Grosse.

This talk is part of the Partial Differential Equations seminar series.