Conformal type of ends of revolution in space forms of constant sectional curvature

TL;DR

This paper investigates the conformal types of complete ends of revolution in space forms, establishing parabolicity in Euclidean and spherical cases and providing conditions for hyperbolic cases.

Contribution

It characterizes the conformal type of ends of revolution in space forms, including new conditions for hyperbolic cases based on their geometric positioning.

Findings

Ends of revolution in Euclidean space are parabolic.

Ends of revolution in spherical space are parabolic.

Hyperbolic ends have conditions for parabolicity based on their position.

Abstract

In this paper we consider the conformal type (parabolicity or non-parabolicity) of complete ends of revolution immersed in simply connected space forms of constant sectional curvature. We show that any complete end of revolution in the $3$-dimensional Euclidean space or in the $3$-dimensional sphere is parabolic. In the case of ends of revolution in the hyperbolic $3$-dimensional space, we find sufficient conditions to attain parabolicity for complete ends of revolution using their relative position to the complete flat surfaces of revolution.