Calculus of Variations and Geometric Measure Theory

N. Ikoma - A. Malchiodi - A. Mondino

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

created by mondino on 01 Jun 2018
modified on 06 Aug 2018


Accepted Paper

Inserted: 1 jun 2018
Last Updated: 6 aug 2018

Journal: Intern. Math. Res. Not. (IMRN)
Year: 2018


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.