Liouville theorem for harmonic maps with potential
Let \(M\), \(N\) be complete Riemannian manifolds, and let \(H\) be a function on \(N\). A harmonic map with potential \(H\) is a map \(u: M\to N\) that is a critical point of the energy functional \[ E_H(u)=\int_M(e(u)-H(u)), \] where \(e\) is the energy density. Assume that \(M\) is a complete noncompact manifold with only one end, and that there is a nonincreasing function \(k\geq 0\) with \(\int_0^\infty t k(t) dt<+\infty\) such that the sectional curvature of \(M\) is bounded below by \(-k(d(\cdot,x))\) for some point \(x\in M\). Let the sectional curvature of \(N\) be bounded above by some \(K\geq 0\), and let \(B_q(\tau)\) be a ball of radius \(\tau<\pi/2\sqrt K\), lying inside the cut locus of \(q\). Finally, assume that \({\partial H\over\partial\rho}\leq 0\), where \(\rho\) denotes the distance from \(q\). The author proves that then any harmonic map \(u: M\to B_q(\tau)\) with potential \(H\) is constant, and that \(u(M)\) is a critical point of \(H\). On the other hand, for \(M\) having more than two ends, the existence of nonconstant harmonic maps \(u: M\to B_q(\tau)\) is known.
scientific article; zbMATH DE number 1158115