Concentration of small Willmore spheres in Riemannian 3-manifolds

Abstract

Given a 3-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 \(\bar{p}\in M\), then \(Scal(\bar{p})=0\) and \(\nabla Scal(\bar{p})=0\) (resp. \(\nabla Scal(\bar{p})=0\)). Moreover, a suitably rescaled sequence smoothly converges, up to subsequences and reparametrizations, to a round sphere in the euclidean 3-dimensional space. This generalizes previous results of Lamm and Metzger contained in \cite{LM1}-\cite{LM2}. An application to the Hawking mass is also established.