Complex length of short curves and Minimal Fibration in hyperbolic $3$-Manifolds fibering over the circle

TL;DR

This paper explores the relationship between complex lengths of curves and minimal surfaces in hyperbolic 3-manifolds, revealing new examples of manifolds with unique minimal surface properties and analyzing geometric conditions.

Contribution

It introduces new results on the existence and non-foliation of minimal surfaces in fibered hyperbolic 3-manifolds based on complex length conditions, supported by explicit computational examples.

Findings

Existence of hyperbolic 3-manifolds not foliated by minimal surfaces.

Construction of quasi-Fuchsian manifolds with many minimal surfaces.

Relation between complex length and minimal surface properties.

Abstract

We investigate the maximal solid tubes around short simple geodesics in hyperbolic three-manifolds and how complex length of curves relate to closed, incompressible, least area minimal surfaces. As applications, we prove, there are some closed hyperbolic three-manifolds fibering over the circle which are not foliated by closed incompressible minimal surfaces diffeomorphic to the fiber. We also show, the existence of quasi-Fuchsian manifolds containing arbitrarily many embedded closed incompressible minimal surfaces. Our strategy is to prove main theorems under natural geometric conditions on the complex length of closed curves on a fibered hyperbolic three-manifold, then we find explicit examples where these conditions are satisfied via computer programs.