Extending Immersions into the Sphere

TL;DR

This paper investigates the conditions under which an immersed circle in the 2-sphere can be extended to an immersion of the disc, analyzing existence and uniqueness through combinatorial methods inspired by Blank's planar case.

Contribution

It introduces a combinatorial framework for understanding extension and uniqueness problems of immersions from circles to discs in the sphere, extending previous planar results.

Findings

Characterization of extension conditions based on combinatorial structures

Criteria for uniqueness of the extension

Extension and uniqueness results specific to the 2-sphere

Abstract

We study the problem to extend an immersed circle f in the 2-dimensional sphere to an immersion of the disc. We analyze existence and uniqueness for this problems in terms of the combinatorial structure of a word assigned to f. Our techniques are based on ideas of Blank who studied the extension problem in case of a planar target.