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2014 Concentration of small Willmore spheres in Riemannian 3-manifolds
Paul Laurain, Andrea Mondino
Anal. PDE 7(8): 1901-1921 (2014). DOI: 10.2140/apde.2014.7.1901

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.

Citation

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Paul Laurain. Andrea Mondino. "Concentration of small Willmore spheres in Riemannian 3-manifolds." Anal. PDE 7 (8) 1901 - 1921, 2014. https://doi.org/10.2140/apde.2014.7.1901

Information

Received: 17 March 2014; Accepted: 4 October 2014; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1322.49071
MathSciNet: MR3318743
Digital Object Identifier: 10.2140/apde.2014.7.1901

Subjects:
Primary: 35J60 , 49Q10 , 53C21 , 53C42 , 83C99

Keywords: blow-up technique , concentration phenomena , fourth-order nonlinear elliptic PDEs , Hawking mass , Willmore functional

Rights: Copyright © 2014 Mathematical Sciences Publishers

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