Concentration of small Willmore spheres in Riemannian 3-manifolds

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.