  We discuss the generalized Plateau problem in the 3+1 dimensional
Schwarzschild background. This represents the physical situation, which could
for instance have appeared in the early universe, where a cosmic membrane (thin
domain wall) is located near a black hole. Considering stationary axially
symmetric membranes, three different membrane-topologies are possible depending
on the boundary conditions at infinity: 2+1 Minkowski topology, 2+1 wormhole
topology and 2+1 black hole topology.
  Interestingly, we find that the different membrane-topologies are connected
via phase transitions of the form first discussed by Choptuik in investigations
of scalar field collapse. More precisely, we find a first order phase
transition (finite mass gap) between wormhole topology and black hole topology;
the intermediate membrane being an unstable wormhole collapsing to a black
hole. Moreover, we find a second order phase transition (no mass gap) between
Minkowski topology and black hole topology; the intermediate membrane being a
naked singularity.
  For the membranes of black hole topology, we find a mass scaling relation
analogous to that originally found by Choptuik. However, in our case the
parameter $p$ is replaced by a 2-vector $\vec{p}$ parametrizing the solutions.
We find that $Mass\propto|\vec{p}-\vec{p}_*|^\gamma$ where $\gamma\approx
0.66$. We also find a periodic wiggle in the scaling relation.
  Our results show that black hole formation as a critical phenomenon is far
more general than expected.
