Immersed spheres of finite total curvature into manifolds

Andrea Mondino; Tristan Rivière

doi:10.1515/acv-2013-0106

Article

Immersed spheres of finite total curvature into manifolds

Andrea Mondino and Tristan Rivière

Published/Copyright: June 26, 2013

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From the journal Advances in Calculus of Variations Volume 7 Issue 4

Abstract

We prove that a sequence of possibly branched, weak immersions of the 2-sphere 𝕊² into an arbitrary compact Riemannian manifold (Mm,h) with uniformly bounded area and uniformly bounded L²-norm of the second fundamental form either collapse to a point or weakly converges as current, modulo extraction of a subsequence, to a Lipschitz mapping of 𝕊² and whose image is made of a connected union of finitely many, possibly branched, weak immersions of 𝕊² with finite total curvature. We prove moreover that if the sequence belongs to a class γ of π2(Mm), the limiting Lipschitz mapping of 𝕊² realizes this class as well.

Keywords: Conformal differential geometry; weak immersions; minimal surfaces; optimal surfaces; geometric PDE; elliptic regularity theory

MSC: 30C70; 58E15; 58E30; 49Q10; 53A30; 35R01; 35J35; 35J48; 35J50

Funding source: ERC

Award Identifier / Grant number: “Geometric Measure Theory in Non-Euclidean Spaces”

This work was written while the first author was visiting the Forschungsinstitut für Mathematik at the ETH Zürich. He would like to thank the Institut for the hospitality and the excellent working conditions.

Received: 2012-3-26

Revised: 2013-6-6

Accepted: 2013-6-17

Published Online: 2013-6-26

Published in Print: 2014-10-1

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Articles in the same Issue

https://doi.org/10.1515/acv-2013-0106

Keywords for this article

Conformal differential geometry; weak immersions; minimal surfaces; optimal surfaces; geometric PDE; elliptic regularity theory