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Let $(M,g)$ be a 3-dimensional Riemannian manifold. The goal of the paper itto show that if $P_{0}\in M$ is a non-degenerate critical point of the scalarcurvature, then a neighborhood of $P_{0}$ is foliated by area-constrainedWillmore spheres. Such a foliation is unique among foliations byarea-constrained Willmore spheres having Willmore energy less than $32\pi$,moreover it is regular in the sense that a suitable rescaling smoothlyconverges to a round sphere in the Euclidean three-dimensional space. We alsoestablish generic multiplicity of foliations and the first multiplicity resultfor area-constrained Willmore spheres with prescribed (small) area in a closedRiemannian manifold. The topic has strict links with the Hawking mass.

Foliation by area-constrained Willmore spheres near a non-degenerate critical point of the scalar curvature