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

Let (M,g) be a three-dimensional Riemannian manifold. The goal of the paper is to show that if P0∈M is a nondegenerate critical point of the scalar curvature, then a neighborhood of P0 is foliated by area-constrained Willmore spheres. Such a foliation is unique among foliations by area-constrained Willmore spheres having Willmore energy less than 32π⁠; moreover, it is regular in the sense that a suitable rescaling smoothly converges to a round sphere in the Euclidean three-dimensional space. We also establish generic multiplicity of foliations and the 1st multiplicity result for area-constrained Willmore spheres with prescribed (small) area in a closed Riemannian manifold. The topic has strict links with the Hawking mass