Erwin Schrödinger Lecture: Hans Werner Ballmann (U Bonn and MPI of Mathematics, Bonn) Small eigenvalues on hyperbolic surfaces


Eigenvalues of the Laplacian on closed hyperbolic surfaces are called small, if they lie below $1/4$, the bottom of the spectrum of the Laplacian on the hyperbolic plane. Buser showed that, for any $\varepsilon>0$, the surface $S$ of genus $g\ge2$ carries a hyperbolic metric such that $\lambda_{2g-3}<\varepsilon$, where the eigenvalues are counted according to their magnitude.
He also showed that $\lambda_{2g-2}\ge c>0$, where $c$ is independent of genus and hyperbolic metric, and conjectured that $c=1/4$ is the best constant. I will discuss the conjecture of Buser, its recent solution by Otal and Rosas, and some related problems and results.

December 3, 2013 at 5 p.m.

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Dec. 3, 2013