  It is proposed that there exists a class of pulsars, called weak pulsars, for
which the large-scale magnetosphere, and hence the gamma-ray emission, are
independent of the detailed pattern of plasma production. The weak pulsar
magnetosphere and its gamma-ray emission are uniquely determined by just three
parameters: spin, dipole, and the spin-dipole angle. We calculate this
supposedly unique pulsar magnetosphere in the axisymmetric case. The
magnetosphere is found to be very close to (although interestingly not fully
identical with) the magnetosphere we have previously calculated, explaining the
phenomenological success of the old calculation.
  We offer only a highly tentative proof of this "Pulsar No-Hair Theorem". Our
analytics, while convincing in its non-triviality, is incomplete, and counts
only as a plausibility argument. Our numerics, while complete, is dubious.
  The plasma flow in the weak pulsar magnetosphere turns out to be even more
intricate than what we have previously proposed: some particles, after being
created near the star, move beyond the light cylinder and then return to the
star.
