Concentration of small Willmore spheres in Riemannian 3-manifolds

Laurain, Paul; Mondino, Andrea

doi:10.2140/apde.2014.7.1901

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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

DOI: 10.2140/apde.2014.7.1901

Abstract

Given a three-dimensional Riemannian manifold $(M, g)$ , we prove that, if $(Φ_{k})$ is a sequence of Willmore spheres (or more generally area-constrained Willmore spheres) having Willmore energy bounded above uniformly strictly by $8 π$ and Hausdorff converging to a point $\bar{p} \in M$ , then $Scal (\bar{p}) = 0$ and $\nabla Scal (\bar{p}) = 0$ (respectively, $\nabla Scal (\bar{p}) = 0$ ). 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

Vol. 7, No. 8, 2014