  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.
