Smooth solution to higher dimensional complex Plateau problem

TL;DR

This paper extends Yau's solution to the complex Plateau problem from hypersurfaces to higher dimensions, proving a conjecture for certain singularities and simplifying the verification process to a single invariant.

Contribution

It generalizes Yau's conjecture to higher dimensions and confirms it for local complete intersection singularities, enabling a simplified approach to the complex Plateau problem.

Findings

Proved Yau's conjecture for dimension n≥3 in local complete intersection singularities.

Solved the complex Plateau problem of hypersurface type in higher dimensions using one invariant.

Extended the understanding of numerical invariants related to complex singularities.

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.