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Perturbations of the harmonic map equation. (English) Zbl 1063.58012

The authors generalize the result of compactness of the space of harmonic maps in any given homotopy class. Let \(M\), \(M'\) be closed manifolds, and suppose \(M'\) has nonpositive sectional curvature. A map \(u\) \(:\) \(M\) \(\rightarrow\) \(M'\) is harmonic if it satisfies the harmonic map equation \(\tau (u)\) \(=\) \(0\), where \(\tau (u)\) is the energy tension field. The authors are concerned with solutions of a semilinear (or quasilinear) perturbed equation \(\tau (u)\) \(+\) \(F(x,\,u(x))\) \(=\) \(0\), where \(F\) is an \(x\)-dependent vector field of class \({C}^k\). Let \(S_F\) denote the totality of classical solutions \(u\) of the perturbed equation, and let \(\zeta\) denote any given homotopy class. The authors prove that \(S_F\) \(\cap\) \(\zeta\) is compact in the \({C}^{k+1}\)-topology. They also give an estimate of a distance between two maps in any same homotopy class using the energy of the maps.

MSC:

58E20 Harmonic maps, etc.
53C43 Differential geometric aspects of harmonic maps
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