Constrained Willmore and CMC tori in the 3-sphere

Constrained Willmore surfaces are critical points of the Willmore functional under conformal variations. As shown in [5] one can associate with any conformally immersed constrained Willmore torus f a compact Riemann surface Σ, such that f can be reconstructed in terms of algebraic data on Σ. Particularly interesting examples of constrained Willmore tori are the tori with constant mean curvature (CMC) in a 3-dimensional space form. It is shown in [11] and in [14] that the spectral curves of these tori are hyperelliptic. In this paper we show under mild conditions that a constrained Willmore torus f   in S3S3 is a CMC torus in a 3-dimensional space form if its spectral curve has the structure of a CMC spectral curve.