## Concentration of small Willmore spheres in Riemannian 3-manifolds.(English)Zbl 1322.49071

Summary: Given a three-dimensional Riemannian manifold $$(M,g)$$, we prove that, if $$(\Phi_k)$$ 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 $$\operatorname{Scal}(\overline{p})=0$$ and $${\nabla \operatorname{Scal}(\overline{p})=0}$$ (respectively, $$\nabla \operatorname{Scal}(\overline{p})=0$$). 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.

### MSC:

 49Q10 Optimization of shapes other than minimal surfaces 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 35J60 Nonlinear elliptic equations 83C99 General relativity
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