Bruce Solomon will carry out research in two fairly independent projects, although the techniques to be employed will be similar in both cases. The first is the study of constant mean curvature surfaces. Most work here has been done on minimal surfaces, the case of zero curvature. In recent years the work of Wente has stimulated interest in the case of non-zero curvature. This arose from Wente's construction of a counterexample to Hopf's long standing conjecture that all soap bubbles are spheres. It is now becoming clear that there is a rich theory of these surfaces analogous but different from that of minimal surfaces. The second area involves the study of isoparametric surfaces. These have the property that the distinct principal curvatures at each point are the same. Interest in such surfaces stems from the fact that they behave in a fashion very similar to the homogeneous spaces. Solomon will focus his attention on the spectral theory of isoparametric minimal surfaces in spheres. The first topic will involve an investigation of the moduli space of complete constant mean curvature surfaces. The intention here is to compare and contrast this theory with that of minimal surfaces. The study of spectral properties of isoparametric surfaces is important since they are in many ways the natural successors to the symmetric spaces and they provide the only other setting in which the spectrum of the Laplace operator is known. The eigenvalues are in fact integers and so it is natural to investigate the geometric consequences of such an observation. The key to the study of these surfaces is the fact that they decompose into mutually orthogonal geodesic foliations by subspheres of the ambient sphere.
