Concentration of small Willmore spheres in Riemannian 3-manifolds. (English) Zbl 1322.49071

Laurain, Paul; Mondino, Andrea

Anal. PDE 7, No. 8, 1901-1921 (2014).

Summary: Given a three-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 \(\overline{p}\in M\), then \(\operatorname{Scal}(\overline{p})=0\) and \({\nabla \operatorname{Scal}(\overline{p})=0}\) (respectively, \(\nabla \operatorname{Scal}(\overline{p})=0\)). Moreover, a suitably rescaled sequence smoothly converges, up to subsequences and reparametrizations, to a round sphere in the Euclidean three-dimensional space. This generalizes previous results of Lamm and Metzger. An application to the Hawking mass is also established.

MSC:

49Q10	Optimization of shapes other than minimal surfaces
53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
35J60	Nonlinear elliptic equations
83C99	General relativity

Keywords:

Willmore functional; Riemannian 3-manifolds; Hawking mass; blow-up technique; concentration phenomena; nonlinear elliptic PDEs