Cannon-Thurston Maps, Harmonic Maps, and Minimal Surfaces in Fibered Hyperbolic 3-Manifolds

Alberto Verjovsky

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arXiv · 2609.35858

Cannon-Thurston Maps, Harmonic Maps, and Minimal Surfaces in Fibered Hyperbolic 3-Manifolds

Alberto Verjovsky

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Abstract

Let $M_f$ be the closed hyperbolic $3$--manifold which is the mapping torus of a pseudo-Anosov homeomorphism $f$ of a closed surface $S$ of genus at least two. The inclusion of the fiber subgroup $π_1(S)\hookrightarrow π_1(M_f)$ has a Cannon--Thurston boundary map $\partialπ_1(S)\cong S^1\to S^2\cong\partialπ_1(M_f)$, which is a Peano curve. We relate this map to two variational representatives of the fiber homotopy class: the harmonic map of Eells--Sampson and Hartman, and a least-area embedded minimal surface. Equivariant lifts of homotopic maps are a bounded distance apart; hence, for every marked conformal structure on $S$, the lift of the harmonic representative extends continuously to the closed disk with boundary value the Cannon--Thurston map, and a least-area fiber lifts to a properly embedded minimal plane in $\Hh^3$ with the same boundary map. We give area and curvature bounds for a least-area fiber, show that its largest principal curvature is at least one, and study the energy of the harmonic representatives as a function on Teichmüller space, its critical points, which are the zeros of the Hopf differential, and its invariance under the monodromy.

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BibTeX RIS

Alberto Verjovsky. 2026-09-25. Cannon-Thurston Maps, Harmonic Maps, and Minimal Surfaces in Fibered Hyperbolic 3-Manifolds. https://arxiv.org/abs/2609.35858

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