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

2025/08/19 by Forstneric, Franc · 1 citation
#32L20 #Complex Variables (math.CV) #FOS: Mathematics #Primary 32W05 #secondary 32Q56

paper · doi:10.48550/arxiv.2508.13660

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 Hölder 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.

Citations

• Runge and Mergelyan theorems on families of open Riemann surfaces
• Oka principle for Levi flat manifolds
• A Parameter Version of Forstnerič's Splitting Lemma

Cited by

• The Oka principle for tame families of Stein manifolds

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• Residual Cauchy-type formula on Riemann surfaces
• H. Cartan's theory for Riemann surfaces
• Generalized Cauchy-Riemann equations and relevant PDE
• Continuous solutions of nonlinear Cauchy-Riemann equations and pseudoholomorphic curves in normal coordinates
• The Transport Oka-Grauert Principle for Simple Surfaces