  We directly compute Wilson loop/Wilson loop correlators on ${\mathbb R}\times
$S$^3$ in AdS/CFT by constructing space-like minimal surfaces that connect two
space-like circular contours on the boundary of global AdS that are separated
by a space-like interval. We compare these minimal surfaces to the disconnected
"double cap" solutions both to regulate the area, and show when the
connected/disconnected solution is preferred. We find that for sufficiently
large Wilson loops no transition occurs because the Wilson loops cannot be
sufficiently separated on the sphere. This may be considered an effect similar
to the Hawking-Page transition: the size of the sphere introduces a new scale
into the problem, and so one can expect phase transitions to depend on this
data. To construct the minimal area solutions, we employ a reduction a la
Arutyunov-Russo-Tseytlin (used by them for spinning strings), and rely on the
integrability of the reduced set of equations to write explicit results.
