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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.

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