  Recent observations that indicate that some extrasolar planets observed in
transit can experience mass loss from their surfaces. Motivated by these
findings, this paper considers outflows from Hot Jupiters in the regime where
the flow is controlled by magnetic fields. Given the mass loss rates estimated
from current observations --- and from theoretical arguments --- magnetic
fields will dominate the flow provided that field strength near the planet is
greater than $\sim1$ gauss, comparable to the surface fields of the Sun and
Jupiter. The problem can be separated into an inner regime, near the planet,
where the outflow is launched, and an outer regime where the flow follows
(primarily) stellar field lines and interacts with the stellar wind. This paper
concentrates on the flow in the inner regime. For a dipole planetary field with
a spatially constant background contribution, we construct a set of orthogonal
coordinates that follow the field lines and determine the corresponding
differential operators. Under the assumption of isothermal flow, we
analytically find the conditions required for escaping material to pass
smoothly through the sonic transition, and then estimate the mass outflow
rates. These magnetically controlled outflows differ significantly from
previous spherical models: The outflow rates are somewhat smaller, typically
${\dot M}$ $\sim 10^{9}$ g/s, and the flow is launched primarily from the polar
regions of the planet. In addition, if the stellar wind is strong enough, the
flow could be reversed and the planet could gain mass from the star.
