Singular points of harmonic maps from 4-dimensional domains into 3-spheres. (English) Zbl 1097.53044

Nakajima, Tôru

Duke Math. J. 132, No. 3, 531-543 (2006).

This paper has two main results:
1) Let \(\Omega \) be a bounded domain in \(\mathbb R^4\) and \(u:\Omega \rightarrow S^{3}\) a stable-stationary harmonic map. Then the mapping degree of \(u\) at a singular point is \(\pm 1\).
2) Any non-constant stable-stationary tangent map from the ball \(B^{4}\) to \(S^{3}\) is a homogeneous extension of an isometry of \(S^{3}\).

Reviewer: R. E. Stong (Charlottesville)

MSC:

53C43	Differential geometric aspects of harmonic maps
58C20	Differentiation theory (Gateaux, Fréchet, etc.) on manifolds

Keywords:

mapping degree; stable-stationary tangent map; isometry

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