  We prove that on a closed surface, for any $c>0$, our min-max theory for
prescribing mean curvature produces a solution given by a curve of constant
geodesic curvature $c$ which is almost embedded, except for finitely many
points, at which the solution is a stationary junction with integer density.
Moreover, each smooth segment has multiplicity one. The key is a classification
of blowups which is new even for $c=0$.
