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I know that the Clifford torus in $S^3$ is Willmore. I can also think about $S^1$-equivariant tori and Hopf tori as Willmore in the $3$-sphere.

Does anyone know if there are more Willmore tori in $S^3$? Any source, idea or comment is very welcome.

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Yes, there are many examples . First of all, there are infinitely many minimal tori (immersed, not embedded ) in the 3-sphere. There are also minimal surfaces with embedded planar ends of genus one. Also, there is the Babich-Bobenko paper in which Willmore tori with umbilical lines are constructed.

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  • $\begingroup$ In case you could also answer this related question that came to my mind: are the minimal tori of revolution in $S^3$ the unique Willmore tori of revolution in $S^3$? $\endgroup$
    – Edu
    Commented Sep 12, 2017 at 13:13
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    $\begingroup$ Yes, willmore tori of revolution are isothermic hence minimal in a space form. This is a result in the thesis of Jörg Richter. All closed examples are minimal in S^3, see for example 'constrained willmore tori and elastic curves' by Lynn Heller. $\endgroup$
    – Sebastian
    Commented Sep 12, 2017 at 18:32

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