Kohn-Rossi cohomology and its application to the complex Plateau problem, III

Abstract

Let $X$ be a compact connected strongly pseudoconvex $CR$ manifold of real dimension $2_{n − 1}$ in $\mathbb{C}^N$. It has been an interesting question to find an intrinsic smoothness criteria for the complex Plateau problem. For $n\ge 3$ 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 $N \ge n + 1$, 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.