  In this paper we study the Liouville type properties for solutions to the
steady incompressible Navier-Stoks equations in $\mathbf{R}^{3}$. It is shown
that any solution to the steady Navier-Stokes equations in $\mathbf{R}^{3}$
with finite Dirichlet integral and vanishing velocity field at far fields must
be trivial. This solves an open problem. The key ingredients of the proof
include a Hodge decomposition of the energy-flux and the observation that the
square of the deformation matrix lies in the local Hardy space. As a
by-product, we also obtain a Liouville type theorem for the steady
density-dependent Navier-Stokes equations.
