On higher dimensional complex Plateau problem

Let $X$ be a compact connected strongly pseudoconvex $CR$ manifold of real dimension $2n-1$ 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 first and third authors introduced 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. For $n\ge 3$ and $N>n+1$, the problem still remains open. In this paper, we generalize the invariant $g^{(1,1)}(X)$ to higher dimension as $g^{({\mathrm{\Lambda }}^{n}1)}(X)$ and show that if $g^{({\mathrm{\Lambda }}^{n}1)}(X)=0$, then the interior has at most finite number of rational singularities. In particular, if $X$ is Calabi–Yau of real dimension $5$, then the vanishing of this invariant is equivalent to give the interior regularity up to normalization.