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Willmore surfaces of revolution bounding two prescribed circles

We consider the family of smooth embedded rotationally symmetric annular type surfaces in R 3 having two concentric circles contained in two parallel planes of R 3 as boundary. Minimising the Willmore functional within this class of surfaces we prove the existence of smooth axisymmetric Willmore surfaces having these circles as boundary. When the radii of the circles tend to zero we prove convergence of these solutions to the round sphere.