  Incompressible Euler flows in narrow domains, in which the horizontal length
scale is much larger than other scales, play an important role in applications,
and their leading-order behavior can be described by the hydrostatic Euler
equations. In this paper, we show that steady solutions of the hydrostatic
Euler equations in an infinite strip strictly away from stagnation must be
shear flows. Furthermore, we prove the existence, uniqueness, and asymptotic
behavior of global steady solutions to the hydrostatic Euler equations in
general nozzles. In terms of stream function formulation, the hydrostatic Euler
equations can be written as a degenerate elliptic equation, for which the
Liouville type theorem in a strip is a consequence of the analysis for the
second order ordinary differential equation (ODE). The analysis on the
associated ODE also helps determine the far field behavior of solutions in
general nozzles, which plays an important role in guaranteeing the equivalence
of stream function formulation. One of the key ingredients for the analysis on
flows in a general nozzle is a new transformation, which combines a change of
variable and an Euler-Lagrange transformation. With the aid of this new
transformation, the solutions in the new coordinates enjoy explicit
representations so that the regularity with respect to the horizontal variable
can be gained in a clear way.
