  In this paper we prove that a capillary minimal surface outside the unit ball
in $\mathbb {R}^3$ with one embedded end and finite total curvature must be
either part of the plane or part of the catenoid. We also prove that a
capillary minimal surface outside the unit ball with one end asymptotic to the
end of the Enneper's surface and finite total curvature cannot exist if the
flux vector vanishes on the first homology calss of the surface. Furthermore,
we prove that a capillary minimal surface outside the convex domain bounded by
several spheres with one embedded end and finite total curvature must be part
of the plane.
