Solving $\bar\partial_b$ on hyperbolic laminations

TL;DR

This paper proves the solvability of the $ar ext{d}_b$ equation for continuous (0,1)-forms with coefficients in high tensor powers of a positive CR line bundle on a compact lamination by Riemann surfaces, extending complex analysis tools to laminated structures.

Contribution

It establishes the existence of solutions to the $ar ext{d}_b$ equation on hyperbolic laminations with positive CR line bundles, a novel extension in the theory of several complex variables.

Findings

Existence of solutions for $ar ext{d}_b u = v$ on hyperbolic laminations.

Solutions are continuous sections in high tensor powers of the line bundle.

The result applies to compact sets laminated by Riemann surfaces with positive CR line bundles.

Abstract

Let $X$ denote a compact set which is laminated by Riemann surfaces. We assume that $X$ carries a positive CR line bundle $ L\rightarrow X$. The main result of the paper is that there exists a positive integer $s$ so that if $v$ is any continuous $(0,1)$ form with coefficients in $L^{\otimes s}$ there exists a continuous section $u$ of $L^{\otimes s}$ solving the equation $\bar\partial_b u=v$.