Cannon-Thurston Maps, Harmonic Maps, and Minimal Surfaces in Fibered Hyperbolic 3-Manifolds
Abstract
Let Mf 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 π1(S)↪ π1(Mf) has a Cannon--Thurston boundary map ∂π1(S)≅ S1→ S2≅∂π1(Mf), 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 Hh3 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.
Related
- Minimal Lagrangian surfaces in the two-dimensional complex hyperbolic quadric via the loop group method
- Harmonic Maps from Punctured Riemann Surfaces to the Hyperbolic Plane with Prescribed Scherk Asymptotics
- Topology, Morse Index, and Area of Minimal Surfaces in Three-Manifolds
- ℝ-equivariant minimal Lagrangian surfaces in the complex quadric Q2 and the complex hyperbolic quadric Q2*
- A surface-group action on real-hyperbolic space with no Cannon--Thurston map
BibTeX
@misc{verjovsky2026cannonthurston,
title = {{Cannon-Thurston Maps, Harmonic Maps, and Minimal Surfaces in Fibered Hyperbolic 3-Manifolds}},
author = {Alberto Verjovsky},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35858},
eprint = {2609.35858},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35858},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cannon-Thurston Maps, Harmonic Maps, and Minimal Surfaces in Fibered Hyperbolic 3-Manifolds
AU - Alberto Verjovsky
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.35858
UR - https://arxiv.org/abs/2609.35858
AB - 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)↪ π_1(M_f) has a Cannon--Thurston boundary map ∂π_1(S)≅ S^1→ S^2≅∂π_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.
ER -