Foliation by Area-constrained Willmore Spheres Near a Nondegenerate Critical Point of the Scalar Curvature

Abstract

AbstractLet $(M,g)$ be a three-dimensional Riemannian manifold. The goal of the paper is to show that if $P_{0}\in M$ is a nondegenerate critical point of the scalar curvature, then a neighborhood of $P_{0}$ is foliated by area-constrained Willmore spheres. Such a foliation is unique among foliations by area-constrained Willmore spheres having Willmore energy less than $32\pi $; 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.