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

Journal article

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.

Authors: Ikoma, N; Malchiodi, A; Mondino, A

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Oxford University Press
• Journal: International Mathematics Research Notices
• Volume: 2020
• Issue: 19
• Pages: 6539–6568
• Publication date: 2018-08-31
• Acceptance date: 2018-08-06
• DOI: 10.1093/imrn/rny203
• EISSN: 1687-0247
• ISSN: 1073-7928
• Language: English
• Keywords: nonlinear fourth order partial differential equations; Lyapunov-Schmidt reduction; Hawking mass; foliation; Willmore functional; FFR