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Pré-Publication, Document De Travail Année : 2022

Concentration of small Willmore spheres in Riemannian 3-manifolds

Résumé

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.

Dates et versions

hal-03869743 , version 1 (24-11-2022)

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Paul Laurain, Andrea Mondino. Concentration of small Willmore spheres in Riemannian 3-manifolds. 2022. ⟨hal-03869743⟩
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