Totally geodesic hyperbolic 3-manifolds in hyperbolic link complements of tori in $S^4$

TL;DR

This paper proves that specific hyperbolic link complements in four-dimensional spheres do not contain closed totally geodesic hyperbolic 3-manifolds, advancing understanding of geometric structures in high-dimensional topology.

Contribution

It establishes the non-existence of certain totally geodesic hyperbolic 3-manifolds within particular hyperbolic link complements in $S^4$, a novel result in geometric topology.

Findings

Certain hyperbolic link complements in $S^4$ lack closed embedded totally geodesic hyperbolic 3-manifolds.

Provides new constraints on the geometric structures possible in high-dimensional link complements.

Enhances understanding of the interplay between hyperbolic geometry and 4-manifold topology.

Abstract

In this paper we prove that certain hyperbolic link complements of $2$-tori in $S^4$ do not contain closed embedded totally geodesic hyperbolic $3$-manifolds.