Regularity for the Harvey-Lawson solutions to the complex Plateau problem (Q1179968)

The classical Plateau problem asks for the odd-dimensional real submanifolds of \(\mathbb{C}^ N\) which are boundaries of complex submanifolds in \(\mathbb{C}^ N\). In Ann. Math., II. Ser. 113, 67-110 (1981; Zbl 0464.32012) the author found smooth solutions for the case of a compact orientable connected CR-manifold \(M\), \(\dim_{\mathbb{R}}M=2n-1\), \(n\geq 3\), in a Stein manifold of complex dimension \(n+1\). The aim of this paper is to solve the problem for a compact connected \((2n-1)\)-dimensional strongly pseudoconvex CR-manifold \(M\) in \(\mathbb{C}^{n+1}\), \(n\geq 2\). This is the boundary of a complex variety \(V\) in the \(C^ \infty\) sense [cf. \textit{F. R. Harvey} and \textit{H. B. Lawson jun.}, Ann. Math., II. Ser. 102, 223-290 (1975; Zbl 0317.32017)]. Letting \(\tau_ i\) be the number of local moduli of \(V\) at the isolated singularity \(p_ i\) of \(V\), the author proves that \(\tau(M)=\sum\tau_ i\) is left invariant by CR- diffeomorphisms. Moreover, \(M\) is a boundary of the complex submanifold \(V\subseteq \mathbb{C}^{n+1}-M\) if and only if \(\tau(M)=0\).