Morse area and Scharlemann-Thompson width for hyperbolic 3-manifolds

TL;DR

This paper establishes a linear relationship between the handle decomposition complexity and the metric complexity based on Morse function level set areas for compact hyperbolic 3-manifolds, linking topological and geometric measures.

Contribution

It introduces a new connection between handle decomposition complexity and metric complexity for hyperbolic 3-manifolds, providing a unified perspective.

Findings

Handle decomposition complexity is linearly related to metric complexity.

The relationship bridges topological and geometric measures in hyperbolic 3-manifolds.

Provides a new tool for analyzing 3-manifold complexity.

Abstract

Scharlemann and Thompson define a numerical complexity for a 3-manifold using handle decompositions of the manifold. We show that for compact hyperbolic 3-manifolds this is linearly related to a definition of metric complexity in terms of the areas of level sets of Morse functions.