Spectral Geometry of Unduloids

Abstract

This paper examines the eigenvalues and eigenfunctions of the Laplace operatorassociated with a set of mathematically defined surfaces which can be producedexperimentally by attaching two equally sized rings to opposite poles of a soapbubble and separating the rings. The shapes produced are called unduloids. Thesecalculations show 1) for a range of ring sizes, as a function of ring separation,the first indexed eigenvalue has a minimum, pointing to an ``optimum'' shape,and 2) given the eigenfunctions in the form of their differential equations anda preference for symmetry, the underlying unduloid geometry may be deduced.