Vol. 7, No. 8, 2014

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Concentration of small Willmore spheres in Riemannian 3-manifolds

Paul Laurain and Andrea Mondino

Vol. 7 (2014), No. 8, 1901–1921
Abstract

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.

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