Embedding closed hyperbolic 3-manifolds in small volume hyperbolic 4-manifolds

TL;DR

This paper investigates the existence of minimal volume hyperbolic 3-manifolds embedded in 4-manifolds, focusing on the presence of totally geodesic submanifolds and their volume constraints.

Contribution

It provides new results on the conditions under which closed embedded hyperbolic 3-manifolds can exist within small volume hyperbolic 4-manifolds.

Findings

Certain closed hyperbolic 3-manifolds cannot embed in small volume hyperbolic 4-manifolds.

Conditions for the existence of totally geodesic submanifolds are characterized.

The study advances understanding of geometric embedding constraints in hyperbolic manifolds.

Abstract

In this paper we study existence and lack thereof of closed embedded orientable co-dimension one totally geodesic submanifolds of minimal volume cusped orientable hyperbolic manifolds.