Handle attachment and the normalized first eigenvalue

Abstract

We show that the first eigenvalue of a closed Riemannian surface normalized by the area can be strictly increased by attaching a cylinder or a cross cap. As a consequence we obtain the existence of maximizing metrics for the normalized first eigenvalue on any closed surface of fixed topological type. Since these metrics are induced by (possibly branched) minimal immersions into spheres, we find new examples of immersed minimal surfaces in spheres.