Kohn-Rossi Cohomology and its application to the Complex Plateau Problem III

TL;DR

This paper introduces a new CR invariant for 2n-1 dimensional strongly pseudoconvex CR manifolds, providing criteria for the regularity of solutions to the complex Plateau problem, especially resolving cases previously open for over 30 years.

Contribution

It defines a novel CR invariant $g^{(1,1)}(X)$ that determines interior regularity of Harvey-Lawson solutions, extending understanding to the case n=2, N=3.

Findings

Vanishing of $g^{(1,1)}(X)$ implies interior regularity.

Provides a solution for the open case n=2, N=3.

Connects CR invariants with complex Plateau problem regularity.

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 problem has been open for over 30 years. In this paper we introduce 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. In case $n=2$ and N=3, the vanishing of this invariant is enough to give the interior regularity.