  Asymptotically flat, time-symmetric, axially symmetric and conformally flat
initial data for vacuum general relativity are studied numerically on $R^3$
with the interior of a standard torus cut out. By the choice of boundary
condition the torus is marginally outer trapped, and thus a surface of minimal
area. Apart from pure scaling the standard tori are parameterized by a radius
$a\in [0,1]$, where $a=0$ corresponds to the limit where the boundary torus
degenerates to a circle and $a=1$ to a torus that touches the axis of symmetry.
Noting that these tori are the orbits of a $U(1)\times U(1)$ conformal isometry
allows for a simple scheme to solve the constraint, involving numerical
solution of only ordinary differential equations.The tori are unstable minimal
surfaces (i.e. only saddle points of the area functional) and thus can not be
apparent horizons, but are always surrounded by an apparent horizon of
spherical topology, which is analyzed in the context of the hoop conjecture and
isoperimetric inequality for black holes.
