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A regularity theorem for harmonic mappings. (English) Zbl 0607.58010
Recently the following result has been proved by M. Giaquinta and J. Souček [Ann. Sc. Norm. Super., Pisa, Cl. Sci., IV. Ser. 12, 81-90 (1985)] and by R. Schoen and K. Uhlenbeck [Invent. Math. 78, 89-100 (1984; Zbl 0555.58011)]. Every energy minimizing weakly harmonic mapping from a domain \(\Omega\) in some Riemannian m-manifold into a hemisphere \(S^ n_+=\{x\in {\mathbb{R}}^{n+1}:| x| \leq 1\), \(x^ n\geq 0\}\) is regular provided \(3\leq m\leq 6\). In this paper the author extends the above result to the case that the target manifold satisfies the following conditions \(J(P_ 0,r_ 0)\) and (*): \(J(P_ 0,r_ 0):\) there exists some \(r_ 0>0\) and a point \(P_ 0\in N\) with the following properties: i) On \(B(P_ 0,r_ 0)=\{P\in N:\) \(dist(P_ 0,P)\leq r_ 0\}\) there exists a normal coordinate system \((u^ 1,...,u^ n)\) such that \(P_ 0=(0,...,0)\), (ii) For any geodesic \(\gamma\) (t) with \(\gamma (0)=P_ 0\) and any Jacobi field Y(t)\(\perp {\dot \gamma}(t)\) and \(Y(0)=0\), the following holds, \[ d\| Y(t)\| /dt>0\quad in\quad B(P_ 0,r_ 0);\quad d\| Y(t)\| /dt=0\quad on\quad \partial B(P_ 0,r_ 0) \] where \(\| Y(t)\| =\sqrt{h_{ij}Y^ iY^ j}\). (*) For any \(P\in \partial B(P_ 0,r_ 0)\), the following inequality holds, \[ \min \{K_ P(\xi_ 1,\xi_ 2): \xi_ 1={\dot \gamma}(1),\quad \xi_ 2\in T_ PN\} \]
\[ \geq \max \{K_ P(\xi_ 1,\xi_ 2): \xi_ 1,\xi_ 2\in T_ PN,\xi_ 1,\xi_ 2\perp {\dot \gamma}(1)\}, \] where \(K_ P(\xi,\eta)\) denotes the sectional curvature of N at P with respect to the plane section spanned by \(\xi\) and \(\eta\), and \(\gamma\) (t) denotes the geodesic joining \(P_ 0\) and P such that \(\gamma (0)=P_ 0\), \(\gamma (1)=P\).

58E20 Harmonic maps, etc.
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