Liouville type theorems of \(f\)-harmonic maps with potential
Two different generalizations of harmonic maps between Riemannian manifolds are put together here. As a variant of \(\int|Du(x)|^2\,dx\), the functional \(\int f(x)|Du(x)|^2\,dx\) can be considered with a positive function \(f\); its stationary points are called \(f\)-harmonic maps. If the domain dimension is not \(2\), this is not really more general, since the factor can be interpreted as the effect of a conformal change of metrics, but nevertheless the paper allows that case. The other variant is that of adding a potential to the functional. This has been done in connection with harmonic maps (the \(f\equiv1\) case), and is generalized here to nonconstant \(f\). The paper combines established methods for both \(f\)-harmonic maps and harmonic maps with potential, and derives standard results like monotonicity formulas and Liouville type theorems.
scientific article; zbMATH DE number 6370028