×

Repulsion and quantization in almost-harmonic maps, and asymptotic of the harmonic map flow. (English) Zbl 1065.58007

Harmonic maps are critical points of the energy functional and are characterised by the vanishing of the tension field.
In the case of Riemann surfaces much is known and harmonic maps between \(2\)-spheres must be rational maps (or their complex conjugates) and their energy a multiple of \(4\pi\). The study of the harmonic flow, i.e. the parabolic equation derived from the tension field, is particularly interesting in this set-up, because the positive curvature of the target prevents the automatic convergence of the flow, which is the angular stone of the Eells-Sampson theorem.
The new phenomenon appearing for the harmonic flow between \(2\)-spheres is the “bubbling effect”, i.e. the existence of a finite number of points where, after suitable scaling, the flow converges to harmonic maps, different from the one obtained away from these points.
As part of the analysis of the harmonic flow, this article studies maps of bounded energy and \(L^{2}\)-small tension field. Such maps must be close, in \(L^{2}\), though not necessarily in \(W^{1,2}\), to a harmonic map, called the body map. Moreover their energy, in general, must be close to \(4 \pi k\) (\(k \in {\mathbb Z}\)), the closeness being bounded by a constant times the \(L^{2}\)-norm of the tension field.
Another remarkable estimate is the bound on the length scale of (some) bubbles developing, in terms of a constant times the exponential of minus the \(L^{2}\)-norm of the tension field.
This has implications for the harmonic map flow, in particular, the energy decreases very slowly and the flow converges uniformly in time.

MSC:

58E20 Harmonic maps, etc.
53C43 Differential geometric aspects of harmonic maps
PDFBibTeX XMLCite
Full Text: DOI