Small eigenvalues on Riemann surfaces of genus 2 (Q1191362)

Let \(M\) be a closed Riemannian surface of genus \(g\geq2\) with a metric of constant curvature \(-1\) and let \(0\leq\lambda_{1}\leq\lambda_{2}...\) be the eigenvalues of the Laplacian where each eigenvalue is repeated according to its multiplicity. An eigenvalue is said to be small if the eigenvalue is less than \(1/4\); \(0\) is a small eigenvalue. It is known that there are surfaces with \(2g-2\) small eigenvalues; the author proved earlier that there are at most \(4g-4\) small eigenvalues. The author conjectured that \(2g-2\) is the appropriate bound; the conjecture is proved for \(g=2\) in this paper.