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.

19 pages

References in corpus (7)

• Willmore Spheres in Compact Riemannian Manifolds
• Immersed Spheres of Finite Total Curvature into Manifolds
• Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds
• Minimizers of the Willmore functional with a small area constraint
• Concentration of CMC Surfaces in a 3-manifold
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Cited by in corpus (4)

• Foliation by area-constrained Willmore spheres near a non-degenerate critical point of the scalar curvature
• Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds I: Minimization
• Some rigidity results for the Hawking mass and a lower bound for the Bartnik capacity
• Local foliations by critical surfaces of the Hawking energy and small sphere limit