  We consider a steady axisymmetric solution of the Euler equations for a fluid
(incompressible and with zero vorticity) with a free surface, acted on only by
gravity. We analyze stagnation points as well as points on the axis of
symmetry. At points on the axis of symmetry which are not stagnation points,
constant velocity motion is the only blow-up profile consistent with the
invariant scaling of the equation. This suggests the presence of downward
pointing cusps at those points.
  At stagnation points on the axis of symmetry, the unique blow-up profile
consistent with the invariant scaling of the equation is Garabedian's pointed
bubble solution with water above air. Thus at stagnation points on the axis of
symmetry with no water above the stagnation point, the invariant scaling of the
equation cannot be the right scaling. A fine analysis of the blow-up velocity
yields that in the case that the surface is described by an injective curve,
the velocity scales almost like $\sqrt{X^2+Y^2+Z^2}$ and is asymptotically
given by the velocity field $$V(\sqrt{X^2+Y^2},Z)=c (-\sqrt{X^2+Y^2}, 2Z)$$
with a nonzero constant $c$.
  The last result relies on a frequency formula in combination with a
concentration compactness result for the axially symmetric Euler equations by
J.-M. Delort. While the concentration compactness result alone does not lead to
strong convergence in general, we prove the convergence to be strong in our
application.
