Explicit harmonic and wave maps into variable-curvature surfaces
Explicit harmonic and wave maps are typically available only in highly symmetric or constant-curvature settings, where additional symmetry or integrability structures are present. We develop a reduction framework for pseudo-Riemannian surfaces that extends explicit constructions to a geometrically significant class of variable-curvature targets. For target metrics of the form $A(R)\,dR^2 - δ^2 B(R)\,dS^2$, a geometrically adapted travelling-wave ansatz reduces the Euler--Lagrange system to a solvable system of first-order ODEs. The method applies simultaneously to harmonic and wave maps, treating the elliptic and hyperbolic regimes uniformly within a single framework. As concrete applications, we construct explicit harmonic maps into ellipsoids, Lorentzian wave maps into hyperboloids and the Schwarzschild exterior, and a mixed-signature example, all in genuinely variable-curvature geometries where explicit constructions are substantially less accessible.