Isospectral surfaces of small genus

Abstract

In this note, we will construct simple examples of isospectral surfaces. In what follows, we will use the term “surface” to mean a surface endowed with a Riemannian metric, while the term “Riemann surface” will be reserved for a surface endowed with a metric of constant curvature. We will show: THEOREM 1. There exist pairs of surfaces S1 and S2 of genus 3, such that S1 and S2 are isospectral but not isometric.THEOREM 2. There exist pairs of Riemann surfaces S1 and S2 of genus 4 and 6, which are isospectral but not isometric.THEOREM 3. There exist unoriented surfaces S1 and S2 of Euler characteristic X(S1) = X(S2) = — 6 which are isospectral but not isometric.