On higher dimensional complex Plateau problem

TL;DR

This paper generalizes a CR invariant to higher dimensions to provide criteria for the smoothness of solutions to the complex Plateau problem, linking invariant vanishing to interior regularity and singularity finiteness.

Contribution

It introduces a new higher-dimensional CR invariant $g^{( abla^n 1)}(X)$ and establishes its vanishing as a condition for interior regularity and limited singularities.

Findings

Vanishing of the invariant implies finite rational singularities.

For Calabi-Yau 5-manifolds, invariant vanishing characterizes interior regularity.

Generalizes previous invariants to higher dimensions.

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.