Smooth solutions to the complex Plateau problem

TL;DR

This paper characterizes smooth solutions to the complex Plateau problem for certain CR manifolds, extending previous work and providing a link-theoretic invariant to distinguish smooth points from singularities.

Contribution

It offers a new characterization of smooth solutions for the complex Plateau problem in specific CR manifolds, completing cases previously only partially solved.

Findings

Complete characterization of smooth solutions for strongly pseudoconvex Calabi--Yau CR manifolds of dimension ≥5.

Determination of a link-theoretic invariant distinguishing smooth points from singularities.

Extension of previous results to the hypersurface case when n=2.

Abstract

Building on work of Du, Gao, and Yau, we give a characterization of smooth solutions, up to normalization, of the complex Plateau problem for strongly pseudoconvex Calabi--Yau CR manifolds of dimension $2n-1 \ge 5$ and in the hypersurface case when $n=2$. The latter case was completely solved by Yau for $n \ge 3$ but only partially solved by Du and Yau for $n=2$. As an application, we determine the existence of a link-theoretic invariant of normal isolated singularities that distinguishes smooth points from singular ones.