Connecting topological Hopf singularities

Hardt, Robert; Rivière, Tristan

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 2, pp. 287-344.

Smooth maps between riemannian manifolds are often not strongly dense in Sobolev classes of finite energy maps, and an energy drop in a limiting sequence of smooth maps often is accompanied by the production (or bubbling) of an associated rectifiable current. For finite 2-energy maps from the 3 ball to the 2 sphere, this phenomenon has been well-studied in works of Bethuel-Brezis-Coron and Giaquinta-Modica-Soucek where a finite mass 1 dimensional rectifiable current occurs whose boundary is the algebraic singular set of the limiting map. The relevant algebraic object here is ${\pi }_2\left(S^2\right)$ which provides both the obstruction to strong approximation by smooth maps and the topological structure to the bubbling set and the singular set. With higher homotopy groups, new phenomena occur. For ${\pi }_3\left(S^2\right)$ and the related case of finite 3-energy maps from the 4 Ball to the 2 sphere, there are examples with bubbled objects that no longer have finite mass. We define a new object, a scan, which generalizes a current but still occurs naturally in bubbling while automatically providing the topological connection between the singularities of the limit map. The bubbled scans, which are found via a new compactness theorem, again enjoy a representation using a finite measure $1$ rectifiable set and an integer density function which is now however only $L^{3/4}$ (rather than $L^1$) integrable.

Classification : 58D15, 58E20, 49Q15

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