Gradient estimates and blow-up analysis for stationary harmonic maps

Fang-Hua Lin

Review from Zentralblatt MATH:

Let $M$ and $N$ be both compact, smooth Riemannian manifolds (with possible nonempty, smooth boundary $

tial M$). The author uses a general analysis on the defect measures and energy concentration sets associated with a weakly convergent sequence of stationary harmonic maps between Riemannian manifolds $M$ and $N$. The first result in this paper is to study the gradient estimates and the compactness of stationary maps in the $H^1$-norm. The author provides a necessary and sufficient conditions for the uniform interior and boundary gradient estimates in terms of the total energy of maps. Secondly, he studies the asymptotic behavior at infinity of stationary harmonic maps from $\bbfR^n$ into a compact Riemannian manifold $N$ with bounded normalized energies. He also shows that if analytic target manifolds do not carry any harmonic $S^2$, then the singular sets of stationary maps are $m\le n-4$ rectifiable. Moreover, the author shows that the well-known theorems of Eells and Sampson, of Hamilton for nonpositively curved targets $N$ and generalizes results of {\it M. Giaquinta} and {\it S. Hildebrandt} [J. Reine Angew. Math. 336, 123-164 (1982; Zbl 0583.49028)].

Reviewed by C.Sung

Keywords: stationary harmonic maps; Riemannian manifolds; gradient estimates

Classification (MSC2000): 58E20

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