  We study the behavior of the Yang-Mills flow for unitary connections on
compact and non-compact oriented surfaces with varying metrics. The flow can be
used to define a one dimensional foliation on the space of SU(2)
representations of a once punctured surface. This foliation universalizes over
Teichm\"uller space and is equivariant with respect to the action of the
mapping class group. It is shown how to extend the foliation as a singular
foliation over the Strebel boundary of Teichm\"uller space, and continuity of
this extension is the main result of the paper.
