Abstract We introduce the natural notion of ( p , q )-harmonic morphisms between Riemannian manifolds. This unifies several theories that have been studied during last decades. then study special case when maps involved are complex-valued. For these we find a characterisation and provide new non-trivial examples in important cases.

We prove versions of various standard inequalities in which the dependence of the constant on the metric is explicit. 1 Technical results Definition. Let (M, g) be a smooth Riemannian n-manifold and let x ∈ M . Given Q > 1, k ∈ N, and p > n, the (Q, k, p)-harmonic radius at x, rH(Q, k, p)(x), is the supremum of reals r such that, on the geodesic ball Bx(r) of center x and radius r, there is a h...