Immersed Spheres of Finite Total Curvature into Manifolds

TL;DR

This paper studies the limits of sequences of weak immersions of the 2-sphere into compact Riemannian manifolds with bounded area and curvature, showing they converge to a union of finite curvature immersions or collapse.

Contribution

It establishes convergence results for sequences of weak immersions with bounded geometric quantities, including the behavior of their limits and the realization of homotopy classes.

Findings

Sequences either collapse or converge to a union of finite curvature immersions.

Limits realize the same homotopy class as the original sequence.

The limiting map is Lipschitz and composed of finitely many immersed spheres.

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.