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.

33 pages. Original preprint (2011). This is the final version to appear in Adv. Calc. Var

References in corpus (5)

• Willmore Spheres in Compact Riemannian Manifolds
• Conformally Invariant Variational Problems
• Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds
• Bubble tree of a class of conformal mappings and applications to Willmore functional
• Energy Quantization for Willmore Surfaces and Applications

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• Existence of generalized totally umbilic 2-spheres in perturbed 3-spheres
• Concentration of small Willmore spheres in Riemannian 3-manifolds
• Foliation by area-constrained Willmore spheres near a non-degenerate critical point of the scalar curvature
• Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds I: Minimization