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

Smooth solution to higher dimensional complex Plateau problem