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Limits of an increasing sequence of Riemann surfaces

Diganta Borah, Prachi Mahajan, Jiju Mammen

Abstract

Let $M$ be a Riemann surface which admits an exhaustion by open subsets $M_j$ each of which is biholomorphic to a fixed domain $Ω\subset \mathbb{C}$. We describe $M$ in terms of $Ω$ under various assumptions on the boundary components of $Ω$.

Limits of an increasing sequence of Riemann surfaces

Abstract

Let be a Riemann surface which admits an exhaustion by open subsets each of which is biholomorphic to a fixed domain . We describe in terms of under various assumptions on the boundary components of .

Paper Structure

This paper contains 14 sections, 5 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some notations:
  4. Some definitions:
  5. Some observations about the union problem:
  6. When $M$ is hyperbolic
  7. An useful result:
  8. Proof of Theorem \ref{['T1']}:
  9. Proof of Theorem \ref{['T1']} (i):
  10. Proof of Theorem \ref{['T1']} (ii):
  11. When $\Omega$ has infinite connectivity
  12. When $\Omega = \Delta \setminus A$, where $A \subset \Delta$ is discrete and infinite:
  13. When $\Omega = \Delta \setminus A$, where $A \subset \Delta$ is a countable union of pairwise disjoint closed discs:
  14. Some explicit examples

Key Result

Theorem 1.1

Assume that in the union problem, $M$ is a hyperbolic Riemann surface.

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 3.1
  • Claim
  • proof : Proof of the claim
  • Proposition 3.2
  • proof : Proof of Proposition
  • Claim
  • proof : Proof of the claim
  • Corollary 5.1
  • ...and 4 more