Rigidity of stable cylinders in three-manifolds

TL;DR

This paper demonstrates that the presence of a specific stable cylinder with bifurcation properties in a three-manifold determines the local structure of the manifold and implies it has infinite volume.

Contribution

It introduces a new stability criterion involving bifurcation phenomena that links stable cylinders to the global geometry of three-manifolds.

Findings

Existence of a stable cylinder implies local manifold structure.

Such cylinders enforce the ambient manifold to have infinite volume.

Bifurcation phenomena are key to the stability and geometric implications.

Abstract

In this paper we show how the existence of a certain stable cylinder determines (locally) the ambient manifold where it is immersed. This cylinder has to verify a {\it bifurcation phenomena}, we make this explicit in the introduction. In particular, the existence of such a stable cylinder implies that the ambient manifold has infinite volume.