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

Ikoma, N
Malchiodi, A
Mondino, A

31 August 2018

## Journal:

International Mathematics Research Notices

## Last Updated:

2021-10-11T15:14:53.607+01:00

## DOI:

10.1093/imrn/rny203

## 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.

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