THE GEOMETRY OF EMBEDDED CONSTANT MEAN CURVATURE TORI IN THE 3-SPHERE VIA INTEGRABLE SYSTEMS

We introduce the moduli space of spectral curves of constant mean curvature cylinders of finite type in the 3-sphere. Moduli space parameters are a hyperelliptic Riemann surface and a meromorphic one form. The subset of spectral curves of mean convex Alexandrov embedded cylinders is explicitly determined. We prove that all embedded cmc tori in the 3-sphere are surfaces of revolution using a combination of integrable systems methods and geometric analysis techniques.