Riemann surfaces out of paper

TL;DR

This paper investigates conditions under which a surface formed from a polygon with infinitely many boundary identifications can be given a unique complex structure, and provides a modulus of continuity for its uniformizing map.

Contribution

It establishes a specific condition ensuring the extension of complex structures and derives a modulus of continuity for the uniformizing map on such surfaces.

Findings

Condition for extending complex structure across boundary

Unique complex structure on the quotient surface

Modulus of continuity for the uniformizing map

Abstract

Let S be a surface obtained from a plane polygon by identifying infinitely many pairs of segments along its boundary. A condition is given under which the complex structure in the interior of the polygon extends uniquely across the quotient of its boundary to make S into a closed Riemann surface. When this condition holds, a modulus of continuity is obtained for a uniformizing map on S.