  Building on work of Du, Gao, and Yau, we give a characterization of smooth
solutions, up to normalization, of the complex Plateau problem for strongly
pseudoconvex Calabi--Yau CR manifolds of dimension $2n-1 \ge 5$ and in the
hypersurface case when $n=2$. The latter case was completely solved by Yau for
$n \ge 3$ but only partially solved by Du and Yau for $n=2$. As an application,
we determine the existence of a link-theoretic invariant of normal isolated
singularities that distinguishes smooth points from singular ones.
