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Minimizers of the Willmore functional with a small area constraint. (English) Zbl 1290.49090
Summary: We show the existence of a smooth spherical surface minimizing the Willmore functional subject to an area constraint in a compact Riemannian three-manifold, provided the area is small enough. Moreover, we partially classify complete surfaces of Willmore type with positive mean curvature in Riemannian three-manifolds.

49Q20 Variational problems in a geometric measure-theoretic setting
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