Let X be a compact connected strongly pseudoconvex CR manifold of real dimension 2n . 1 in CN. It has been an interesting
question to find an intrinsic smoothness criteria for the complex Plateau problem. For n 3 and N = n+1, Yau found a necessary
and sufficient condition for the interior regularity of the HarveyLawson solution to the complex Plateau problem by means of
Kohn–Rossi cohomology groups on X in 1981. For n = 2 and N n + 1, the problem has been open for over 30 years. In
this paper we introduce a new CR invariant g(1,1)(X) of X. The vanishing of this invariant will give the interior regularity of the Harvey–Lawson solution up to normalization. In the case n = 2 and N = 3, the vanishing of this invariant is enough to give the interior regularity.