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A compactness theorem for harmonic maps. (English) Zbl 1005.58007
Let $$M$$, $$N$$ be compact Riemannian manifolds. A map $$u$$ of $$M$$ to $$N$$ is called a harmonic map if $$u$$ is smooth and a critical point of the energy functional. In case, where the dimension $$m$$ of $$M$$ is greater then or equal to $$3$$, the author shows that any set of harmonic maps with the uniformly bounded $$m$$-energy is compact in $$C^\infty (M,N)$$. As a corollary he obtains the gradient estimate of harmonic maps.

MSC:
 5.8e+21 Harmonic maps, etc.