#### Vol. 7, No. 8, 2014

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Concentration of small Willmore spheres in Riemannian 3-manifolds

### Paul Laurain and Andrea Mondino

Vol. 7 (2014), No. 8, 1901–1921
##### Abstract

Given a three-dimensional Riemannian manifold $\left(M,g\right)$, we prove that, if $\left({\Phi }_{k}\right)$ is a sequence of Willmore spheres (or more generally area-constrained Willmore spheres) having Willmore energy bounded above uniformly strictly by $8\pi$ and Hausdorff converging to a point $\overline{p}\in M$, then $Scal\left(\overline{p}\right)=0$ and $\nabla Scal\left(\overline{p}\right)=0$ (respectively, $\nabla Scal\left(\overline{p}\right)=0$). Moreover, a suitably rescaled sequence smoothly converges, up to subsequences and reparametrizations, to a round sphere in the euclidean three-dimensional space. This generalizes previous results of Lamm and Metzger. An application to the Hawking mass is also established.

##### Keywords
Willmore functional, Hawking mass, blow-up technique, concentration phenomena, fourth-order nonlinear elliptic PDEs
##### Mathematical Subject Classification 2010
Primary: 49Q10, 53C21, 53C42, 35J60, 83C99
##### Milestones
Accepted: 4 October 2014
Published: 5 February 2015
##### Authors
 Paul Laurain Institut de Mathématiques de Jussieu Paris VII Bátiment Sophie Germain Case 7012 75205 Paris Cedex 13 France Andrea Mondino Department of Mathematics ETH Rämistrasse 101 CH-8092 Zürich Switzerland