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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 $$\gamma (0)=P_ 0$$, $$\gamma (1)=P$$.

##### MSC:
 5.8e+21 Harmonic maps, etc.
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