Concentration of small Willmore spheres in Riemannian 3-manifolds

Abstract

Given a 3-dimensional Riemannian manifold (M,g) , we prove that if (Φ k ) is a sequence of Willmore spheres (or more generally area-constrained Willmore spheres), having Willmore energy bounded above uniformly strictly by 8π , and Hausdorff converging to a point p ¯ ∈M , then Scal(p ¯ )=0 and ∇Scal(p ¯ )=0 (resp. ∇Scal(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.