Kohn–rossi cohomology and its application to the complex plateau problem, III

Let X be a compact connected strongly pseudoconvex CR man- ifold of real dimension 2n − 1 in C^N. 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 Harvey– Lawson 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.