`Small Eigenvalues of Riemannian surfaces'

Abstract:

Any eigenvalue of the Laplace operator, acting on the function space of a hyperbolic surface $S$, below $1/4$ is called a {\it small or exceptional} eigenvalue of $S$. 
Historically, Selberg’s $1/4$-conjecture was a motivation for the study of these eigenvalues. 
Existence of hyperbolic surfaces having small eigenvalues was first obtained by B. Randol (1974, \cite{Ra}). 
Later P. Buser found a simpler construction of such surfaces (1977, \cite{Bu}).
He also found initial bounds on the number of  small eigenvalues of a given surface depending on the topology of the surface (1977, \cite{Bu}). 

Later P. Schumtz (1991, \cite{Sch}) sharpened the methods developed by Buser and from his bounds he (and later Buser also) conjectured that the number of these eigenvalues of a closed hyperbolic surface is at most the Euler characteristic of the surface. An extended version of this conjecture was proved by Otal and Rosas (2009, \cite{OR}). In their paper Otal-Rosas asked if their result can be extended to all smooth surfaces. In a series of three papers \cite{BMM1}, \cite{BMM2} and \cite{BMM3}, joint with Werner Ballmann and Henrik Matthiesen, we have answered an extended version of this last question in the affirmative. In this talk I shall present a short survey of these developments and results.

\bibitem[BMM1]{BMM1} W. Ballmann, H. Matthiesen, S. Mondal, Small eigenvalues of closed surfaces.
\emph{J. Differential Geom.} 103 (2016), no. 1, 1–13, MR3488128, Zbl 1341.53066.


\bibitem[BMM2]{BMM2} W. Ballmann, H. Matthiesen, S. Mondal, Small eigenvalues of surfaces of finite
type. \emph{Compositio Math.} 153 (2017), 1747–1768, MR, Zbl.
\bibitem[BMM3]{BMM3} W. Ballmann, H. Matthiesen, S. Mondal, On the analytic systole of Riemannian
surfaces of finite type. (submitted).

\bibitem[Bu]{Bu} P. Buser, Geometry and spectra of compact Riemann surfaces. Reprint of the
1992 edition. Modern Birkhäuser Classics. Birkhäuser, 2010. xvi+454 pp.,
MR2742784, Zbl 1239.32001.


\bibitem[OR]{OR}J.-P. Otal, E. Rosas, Pour toute surface hyperbolique de genre g, $\lambda_{2g-2} > 1/4.$
\emph{Duke Math. J.} 150 (2009), no. 1, 101–115, MR2560109, Zbl 1179.30041.


\bibitem[Ra]{Ra} B. Randol Small eigenvalues of the Laplace operator on compact Riemann
surfaces. \emph{Bull. Amer. Math. Soc.} 80 (1974), 996–1000.

\bibitem[Sch]{Sch} P. Schmutz, Small eigenvalues on Riemann surfaces of genus 2. \emph{Invent. Math.}
106 (1991), no. 1, 121–138, MR1123377, Zbl 0764.53035.