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arXiv · 1305.6205

Immersed Spheres of Finite Total Curvature into Manifolds

Andrea Mondino·Tristan Rivière
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Abstract

We prove that a sequence of possibly branched, weak immersions of the two-sphere $S^2$ into an arbitrary compact riemannian manifold $(M^m,h)$ with uniformly bounded area and uniformly bounded $L^2-$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 $S^2$ and whose image is made of a connected union of finitely many, possibly branched, weak immersions of $S^2$ with finite total curvature. We prove moreover that if the sequence belongs to a class $γ$ of $π_2(M^m)$ the limiting lipschitz mapping of $S^2$ realizes this class as well.

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Andrea Mondino, Tristan Rivière. 2013-05-27. Immersed Spheres of Finite Total Curvature into Manifolds. https://doi.org/10.1515/acv-2013-0106

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