It is known from general relativity that axisymmetric stationary black holes can be reduced to axisymmetric harmonic maps into the hyperbolic plane $H^2$, while in the Riemannian setting, 4d Ricci-flat metrics with torus symmetry can also be locally reduced to such harmonic maps satisfying a tameness condition. We study such harmonic maps. Applications include a construction of infinitely many new complete, asymptotically flat, Ricci-flat 4-manifolds with arbitrarily large $b_2$. Joint work with Song Sun.