The nonhomogeneous Cauchy-Riemann equation on families of open Riemann surfaces

TL;DR

This paper develops an optimal solution to the nonhomogeneous Cauchy-Riemann equation for families of open Riemann surfaces using the Beltrami equation, with applications to the Oka-Grauert principle.

Contribution

It introduces a method to solve the nonhomogeneous Cauchy-Riemann equation on families of surfaces, extending complex analysis tools to variable structures.

Findings

Provides an optimal solution to the nonhomogeneous Cauchy-Riemann equation.

Establishes the Oka-Grauert principle for complex line bundles on families of open Riemann surfaces.

Abstract

In this paper we use the nonhomogeneous Beltrami equation to give an optimal solution to the nonhomogeneous Cauchy-Riemann equation for continuous or smooth families of complex structures and $(0,1)$-forms of a H\"older class on a smooth open orientable surface. As an application, we obtain the Oka-Grauert principle for complex line bundles on families of open Riemann surfaces.