Elasticae with constant slant in the complex projective plane and new examples of Willmore tori in five spheres

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We exhibit a reduction of variables criterion for the Willmore variational problem. KIt can be considered as an application of the Palais principle of symmetric criticality. Thus, via the Hopf map, we reduce the problem of finding Willmore tori (with a certain degree of symmetry) in the five sphere equipped with its standard conformal structure, to that for closed elasticae in the complex projective plane. Then, we succeed in obtaining the complete classification of elasticae with constant slant in this space. It essentially consists in three kinds of elasticae. Two of them correspond with torsion free elasticae. They lie into certain totally geodesic surfaces of the complex projective plane and their slants reach the extremal values. The third type gives a two-parameter family of helices, lying fully in this space. A nice closure condition, involving the rationality of one parameter, is obtained five sphere. They are Hopf map liftings of the above mentioned families of elasticae. The method also works for a one-parameter family of conformal structures on the five sphere, which defines a canonical deformation of the standard one.
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