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We develop a general Minmax procedure in Euclidian spaces for constructingWillmore surfaces of non zero indices. We implement this procedure to theWillmore Minmax Sphere Eversion in the 3 dimensional euclidian space. Wecompute the cost of the Sphere eversion in terms of Willmore energies ofWillmore Spheres in ${\R}^3$

Willmore Minmax Surfaces and the Cost of the Sphere Eversion