Gradient estimates and blow-up analysis for stationary harmonic maps. (English) Zbl 0949.58017

Lin, Fang-Hua

Ann. Math. (2) 149, No. 3, 785-829 (1999).

Let \(M\) and \(N\) be both compact, smooth Riemannian manifolds (with possible nonempty, smooth boundary \(\partial M\)). 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 \(\mathbb{R}^n\) 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 \(m\leq n-4\) 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 M. Giaquinta and S. Hildebrandt [J. Reine Angew. Math. 336, 123-164 (1982; Zbl 0508.58015)] and R. M. Schoen [Publ., Math. Sci. Res. Inst. 2, 321-358 (1984; Zbl 0551.58011)].
For related results see J. Sacks and K. Uhlenbeck [Ann. Math., II. Ser. 113, 1-24 (1981: Zbl 0462.58014)], R. Schoen and K. Uhlenbeck [J. Differ. Geom. 18, 253-268 (1983; Zbl 0547.58020)], M. Giaquinta and E. Giusti [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 11, 45-55 (1984; Zbl 0543.49018)], D. Preiss [Ann. Math., II. Ser. 125, 537-643 (1987; Zbl 0627.28008)] and L. Simon [Lect. Notes Math. 1161, 206-277 (1985; Zbl 0583.49028)].

Reviewer: C.Sung

MSC:

58E20

Harmonic maps, etc.

Keywords:

stationary harmonic maps; Riemannian manifolds; gradient estimates

Citations:

Zbl 0508.58015; Zbl 0551.58011; Zbl 0462.58014; Zbl 0547.58020; Zbl 0543.49018; Zbl 0627.28008; Zbl 0583.49029; Zbl 0583.49028