A criterion of convergence in the augmented Teichmueller space

TL;DR

This paper establishes a convergence criterion in the augmented Teichmueller space, linking hyperbolic metric convergence with quasiconformal convergence away from nodes, enhancing understanding of geometric limits.

Contribution

It introduces a new criterion for convergence in the augmented Teichmueller space based on hyperbolic and quasiconformal convergence.

Findings

Convergence can be characterized via hyperbolic metrics.

Quasiconformal convergence away from nodes is equivalent.

Provides a practical criterion for geometric analysis.

Abstract

We prove a criterion of convergence in the augmented Teichmueller space that can be phrased in terms of convergence of the hyperbolic metrics or of quasiconformal convergence away from the nodes.