On higher dimensional complex Plateau problem

Abstract

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 $$g^{(\Lambda ^n 1)}(X)$$ and show that if $$g^{(\Lambda ^n 1)}(X)=0$$ , 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.