Immersed Spheres of Finite Total Curvature into Manifolds
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 $\gamma$ of $\pi_2(M^m)$ the limiting lipschitz mapping of $S^2$ realizes this class as well.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2013
- DOI:
- 10.48550/arXiv.1305.6205
- arXiv:
- arXiv:1305.6205
- Bibcode:
- 2013arXiv1305.6205M
- Keywords:
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- Mathematics - Differential Geometry;
- Mathematics - Analysis of PDEs;
- Mathematics - Geometric Topology
- E-Print:
- 33 pages. Original preprint (2011). This is the final version to appear in Adv. Calc. Var