Sharp eigenvalue estimates on degenerating surfaces

We consider the first non-zero eigenvalue ${\lambda }_1$ of the Laplacian on hyperbolic surfaces for which one disconnecting collar degenerates and prove that $8\pi \mathrm{\nabla }\log ({\lambda }_1)$ 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 and Burger to obtain estimates with optimal error rates and obtain new information on the leading order terms of the polyhomogeneous expansion of ${\lambda }_1$ of Albin, Rochon and Sher.