Minimizer of the Willmore energy with prescribed rectangular conformal class

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In this project we aim at determining all tori in 3-space minimizing the Willmore functional with prescribed rectangular conformal type by combining methods from Integrable Systems Theory and Global Analysis. Candidates are given by the real analytic family of constrained Willmore tori starting at the Clifford torus following the homogenous tori, bifurcating at the first bifurcation point to the 2-lobed Delaunay tori and converging to the doubly covered sphere as the conformal type degenerates. Generalization of the conjecture to an open neighborhood of the rectangular conformal class will be investigated. In particular, appropriate candidates in these conformal classes will be constructed, as of now only those in a small neighborhood of the Clifford torus are known.