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Nonlinear Riemann-Hilbert problem for bordered Riemann surfaces

Miran Cerne

TL;DR

The paper extends the nonlinear Riemann-Hilbert problem to bordered Riemann surfaces, proving that for a $C^{k+1}$ family of Jordan curves $\{\gamma_z\}$ in $\mathbb{C}$ containing 0, there exists a holomorphic $f$ on a bordered surface $\Sigma$ with at most $2g+m-1$ zeros such that $f(z)\in\gamma_z$ on the boundary. The construction combines approximate disc-solving with a dbar-gluing approach, a Gromov compactness argument, and a Begehr-Efendiev implicit-function theorem to perturb to an exact solution, while carefully controlling the zero set via index considerations. The results yield divisor-related corollaries and embedding consequences for finitely connected planar domains, and the Appendix furnishes the necessary technical tools, including plurisubharmonic constructions, area-normalization arguments, and sharp a priori estimates. Overall, the work provides a rigorous nonlinear Riemann-Hilbert framework for bordered Riemann surfaces and connects to polynomial hull descriptions and embedding problems in several complex variables.

Abstract

Let S be a bordered Riemann surface with genus g and m boundary components. For a smooth family of smooth Jordan curves in the complex plane parametrized by the boundary of S and such that all curves contain 0 in their interior we show that there exists a holomorphic solution of the corresponding Riemann-Hilbert problem with at most 2g+m-1 zeros on S.

Nonlinear Riemann-Hilbert problem for bordered Riemann surfaces

TL;DR

The paper extends the nonlinear Riemann-Hilbert problem to bordered Riemann surfaces, proving that for a family of Jordan curves in containing 0, there exists a holomorphic on a bordered surface with at most zeros such that on the boundary. The construction combines approximate disc-solving with a dbar-gluing approach, a Gromov compactness argument, and a Begehr-Efendiev implicit-function theorem to perturb to an exact solution, while carefully controlling the zero set via index considerations. The results yield divisor-related corollaries and embedding consequences for finitely connected planar domains, and the Appendix furnishes the necessary technical tools, including plurisubharmonic constructions, area-normalization arguments, and sharp a priori estimates. Overall, the work provides a rigorous nonlinear Riemann-Hilbert framework for bordered Riemann surfaces and connects to polynomial hull descriptions and embedding problems in several complex variables.

Abstract

Let S be a bordered Riemann surface with genus g and m boundary components. For a smooth family of smooth Jordan curves in the complex plane parametrized by the boundary of S and such that all curves contain 0 in their interior we show that there exists a holomorphic solution of the corresponding Riemann-Hilbert problem with at most 2g+m-1 zeros on S.

Paper Structure

This paper contains 4 sections, 14 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. The Riemann-Hilbert problem
  3. Example
  4. Appendix

Key Result

Theorem 1.1

Let ${\rm\Sigma}$ be a bordered Riemann surface with genus $g$ and $m$ real analytic boundary components. Let $\lbrace\gamma_{z}\rbrace_{z\in\partial{\rm\Sigma}}$ be a $C^{k+1}$$(k\ge 3)$ family of Jordan curves in ${\mathbb C}$ which all contain the point $0$ in their interior. Then there exists a

Theorems & Definitions (32)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3: Begehr-Efendiev
  • proof
  • Corollary 2.4
  • ...and 22 more