Minimal Fibrations of Hyperbolic 3-manifolds

TL;DR

The paper addresses whether hyperbolic 3-manifolds that fiber over $S^1$ can admit fibrations by minimal surfaces. By constructing bundles with arbitrarily short geodesics and analyzing cusp geometriy, it establishes that many such manifolds admit no minimal fibrations, and in fact no fibrations by $(\mu,\lambda)$-quasi-area-minimizing fibers for any fixed $0 \le \mu < 1$ and $\lambda \ge 1$ when the geodesics become sufficiently short. The main approach combines Thurston’s Dehn surgery, geometric convergence to cusped limits, and precise cusp-area estimates to obstruct minimal-like foliations. These results illuminate intrinsic obstructions to taut/minimal fibrations in hyperbolic 3-manifolds, linking cusp geometry to global foliation properties with potential implications for Sullivan’s taut foliation framework. The work also poses several natural open questions about the existence of minimal or mean-curvature-bounded fibrations in closed or negatively curved settings.

Abstract

There are hyperbolic 3-manifolds that fiber over the circle but that do not admit fibrations by minimal surfaces. Furthermore these manifolds do not admit fibrations by surfaces that are even approximately minimal.

Figures (2)

• Figure 1: A minimal disk bounded by a curve of hyperbolic length $l$ that lies on the horosphere $\{ z= s \}$ is contained inside a horoball of height $h=s + ( sl^2)/4$. Indicated distances are Euclidean.
• Figure 2: Geometric convergence implies that $K_s$, the compact submanifold of $M$ below a fixed horotorus $T_s$, increasingly resembles the submanifold of $M_j$ below $f_j(T_s)$

Theorems & Definitions (18)

• Theorem 1.1
• Corollary 1.1
• proof
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4