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.


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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