# Negatively Curved Three-Manifolds, Hyperbolic Metrics, Isometric Embeddings In Minkowski Space And The Cross Curvature Flow

## TL;DR

This paper explores the properties of negatively curved three-manifolds, focusing on isometric embeddings into Minkowski space and the use of Cross Curvature Flow to analyze hyperbolic metrics, providing new insights into their rigidity and integrability.

## Contribution

It offers an expository review of rigidity and embedding properties, and establishes a link between integrability and embeddability in the context of the Cross Curvature Flow.

## Key findings

- Solutions with fixed Einstein volume are exactly the integrable solutions.
- Provides insights into rigidity properties related to Minkowski space embeddings.
- Answers a question by Chow and Hamilton regarding the relationship between integrability and embeddability.

## Abstract

This short note is a mostly expository article examining negatively curved three-manifolds. We look at some rigidity properties related to isometric embeddings into Minkowski space. We also review the Cross Curvature Flow (XCF) as a tool to study the space of negatively curved metrics on hyperbolic three-manifolds, the largest and least understood class of model geometries in Thurston's Geometrisation. The relationship between integrability and embedability yields interesting insights, and we show that solutions with fixed Einstein volume are precisely the integrable solutions, answering a question posed by Chow and Hamilton when they introduced the XCF.