Smooth solution to higher dimensional complex Plateau problem

Abstract

Let $X$ be a compact connected strongly pseudoconvex $CR$ manifold of real dimension $2n-1$ in $\mathbb{C}^{N}$. For $n\ge 3$, Yau solved the complex Plateau problem of hypersurface type by checking a bunch of Kohn-Rossi cohomology groups in 1981. In this paper, we generalize Yau's conjecture on some numerical invariant of every isolated surface singularity defined by Yau and the author to any dimension and prove that the conjecture is true for local complete intersection singularities of dimension $n\ge 3$. As a direct application, we solved complex Plateau problem of hypersurface type for any dimension $n\ge 3$ by checking only one numerical invariant.