Constant boundary-value problems for pp-harmonic maps with potential

In this paper, after introducing a large class of manifolds which includes the manifolds with strictly negative curvature bounded between two negative constants as special cases, we study the constant boundary-value problems of pp-harmonic maps with potential defined on such a class of manifolds, and obtain a Liouville-type theorem. The main theorem generalizes that of Karcher and Wood [H. Karcher, J.C. Wood, Non-existence results and growth properties for harmonic maps and forms, J. Reine. Angew. Math. 353 (1984) 165–180] and Chen [Q. Chen, Stability and constant boundary-value problems of harmonic maps with potential, J. Aust. Math. Soc. (Series A) 68 (2000) 145–154] even for the case of the usual harmonic maps or harmonic maps with potential. It can also be applied to the static Landau–Lifshitz equations. Then, using the technique developed there, we prove a Liouville theorem for pp-harmonic maps with finite pp-energy or slowly divergent pp-energy, which answers partially Sampson’s conjecture in a more general case.