Vol. 7, No. 8, 2014

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ISSN: 1948-206X (e-only)
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Concentration of small Willmore spheres in Riemannian 3-manifolds

Paul Laurain and Andrea Mondino

Vol. 7 (2014), No. 8, 1901–1921

Given a three-dimensional Riemannian manifold (M,g), we prove that, if (Φk) 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 (respectively, Scal(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.

Willmore functional, Hawking mass, blow-up technique, concentration phenomena, fourth-order nonlinear elliptic PDEs
Mathematical Subject Classification 2010
Primary: 49Q10, 53C21, 53C42, 35J60, 83C99
Received: 17 March 2014
Accepted: 4 October 2014
Published: 5 February 2015
Paul Laurain
Institut de Mathématiques de Jussieu
Paris VII
Bátiment Sophie Germain
Case 7012
75205 Paris Cedex 13
Andrea Mondino
Department of Mathematics
Rämistrasse 101
CH-8092 Zürich