  We establish global well-posedness and scattering for wave maps from
$d$-dimensional hyperbolic space into Riemannian manifolds of bounded geometry
for initial data that is small in the critical Sobolev space for $d \geq 4$.
The main theorem is proved using the moving frame approach introduced by Shatah
and Struwe. However, rather than imposing the Coulomb gauge we formulate the
wave maps problem in Tao's caloric gauge, which is constructed using the
harmonic map heat flow. In this setting the caloric gauge has the remarkable
property that the main `gauged' dynamic equations reduce to a system of
nonlinear scalar wave equations on $\mathbb{H}^{d}$ that are amenable to
Strichartz estimates rather than tensorial wave equations (which arise in other
gauges such as the Coulomb gauge) for which useful dispersive estimates are not
known. This last point makes the heat flow approach crucial in the context of
wave maps on curved domains.
