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

TL;DR

This paper proves that near a non-degenerate critical point of scalar curvature in a 3D manifold, there exists a unique foliation by area-constrained Willmore spheres, with results on multiplicity and connections to Hawking mass.

Contribution

It establishes the existence, uniqueness, and regularity of foliations by area-constrained Willmore spheres near critical points of scalar curvature, including multiplicity results.

Findings

Existence of foliation near critical points

Uniqueness of foliation with energy less than 32π

Multiplicities of foliations and spheres with prescribed small area

Abstract

Let $(M,g)$ be a 3-dimensional Riemannian manifold. The goal of the paper it to show that if $P_{0}\in M$ is a non-degenerate 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 first 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.