Finite parts of the spectrum of a Riemann surface

Abstract

Let S and $S'$ be compact Riemann surfaces of the same genus g (g$ge 2)$ endowed with the Poincare metric of constant negative curvature -1. par The authors show that for every $epsilon >0$, there exists an integer $m=m(g,epsilon)$ with the property: Assume that (1) injectivity radius of S and $S'ge epsilon$, and (2) the first $m=m(g,epsilon)$ eigenvalues of the Laplacian of S and $S'$ coincide, then S and $S'$ are isospectral. par The authors conjecture that the integer m will not depend on $epsilon$ that is, the theorem holds with an integer which depends only on the genus. par For the proof, a model of Teichmuller space is described and the analyticity of the resolvent of Laplacian on this space is proved. Also the authors note that the injectivity radius is estimated by a finite part of the spectrum $(=$ the number of eigenvalues of the Laplacian in the interval [1/4,1]).