Mean-convex Alexandrov embedded constant mean curvature tori in the 3-sphere

We introduce the moduli space of spectral curves of constant mean curvature (CMC) cylinders of finite type in the round unit 3-sphere. The subset of spectral curves of mean-convex Alexandrov embedded cylinders is explicitly determined using a combination of integrable systems and geometric analysis techniques. We prove that these cylinders are surfaces of revolution. As a consequence, all mean-convex Alexandrov embedded CMC tori in the 3-sphere are surfaces of revolution.