A compactness theorem for harmonic maps (Q1979118)

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.