Yang-Mills moduli space in the adiabatic limit

We consider the Yang-Mills equations for a matrix gauge group $G$ inside the future light cone of 4-dimensional Minkowski space, which can be viewed as a Lorentzian cone $C(H^3)$ over the 3-dimensional hyperbolic space $H^3$. Using the conformal equivalence of $C(H^3)$ and the cylinder $R\times H^3$, we show that, in the adiabatic limit when the metric on $H^3$ is scaled down, classical Yang-Mills dynamics is described by geodesic motion in the infinite-dimensional group manifold $C^{\mathrm{\infty }}(S_{\mathrm{\infty }}^2,G)$ of smooth maps from the boundary 2-sphere $S_{\mathrm{\infty }}^2=\mathrm{\partial }{H}^3$ into the gauge group $G$.