Calculus of Variations and Geometric Measure Theory

P. Laurain - A. Mondino

Concentration of small Willmore spheres in Riemannian 3-manifolds

created by mondino on 26 Oct 2013
modified on 23 Feb 2015


Published Paper

Inserted: 26 oct 2013
Last Updated: 23 feb 2015

Journal: Analysis and PDEs
Year: 2013


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.