Sharp mean-width and Jacobian bounds for Euclidean and hyperbolic harmonic maps
Abstract
We prove a sharp mean-width inequality for monotone zonal operators and derive global and differential bounds for Euclidean and hyperbolic-harmonic self-maps of the unit ball. Let k:[0,π]→ℝ be continuous and nonincreasing, and let Tk be the associated zonal integral operator, (TkF)(ξ)=∫𝕊n−1 k (arccos⟨ξ,η⟩) F(η) dσ(η), where σ is normalized surface measure. For every measurable F:𝕊n−1→𝔹n̅, we prove w (coTkF(𝕊n−1)) ≤2λ1(k), where w denotes mean width, normalized by w(𝔹n̅)=2, and λ1(k) is the eigenvalue of Tk on the space of spherical harmonics of degree one. For strictly decreasing kernels, equality holds exactly for F(η)=Qη almost everywhere, with Q∈ O(n). For the ordinary Poisson kernel, the multiplier is r, yielding sharp mean-width, intrinsic-volume, and image-volume contraction. In particular, |f(r𝔹n)|≤ωn rn without injectivity assumptions, answering the area and higher-dimensional volume question of Koh and Kovalev.
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BibTeX
@misc{zhong2026sharp,
title = {{Sharp mean-width and Jacobian bounds for Euclidean and hyperbolic harmonic maps}},
author = {Deguang Zhong and David Kalaj},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09316},
eprint = {2609.09316},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09316},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp mean-width and Jacobian bounds for Euclidean and hyperbolic harmonic maps
AU - Deguang Zhong
AU - David Kalaj
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09316
UR - https://arxiv.org/abs/2609.09316
AB - We prove a sharp mean-width inequality for monotone zonal operators and derive global and differential bounds for Euclidean and hyperbolic-harmonic self-maps of the unit ball. Let k:[0,π]→ℝ be continuous and nonincreasing, and let T_k be the associated zonal integral operator, (T_kF)(ξ)=∫_𝕊^n−1 k (arccos⟨ξ,η⟩) F(η) dσ(η), where σ is normalized surface measure. For every measurable F:𝕊^n−1→𝔹^n̅, we prove w (coT_kF(𝕊^n−1)) ≤2λ_1(k), where w denotes mean width, normalized by w(𝔹^n̅)=2, and λ_1(k) is the eigenvalue of T_k on the space of spherical harmonics of degree one. For strictly decreasing kernels, equality holds exactly for F(η)=Qη almost everywhere, with Q∈ O(n). For the ordinary Poisson kernel, the multiplier is r, yielding sharp mean-width, intrinsic-volume, and image-volume contraction. In particular, |f(r𝔹^n)|≤ω_n r^n without injectivity assumptions, answering the area and higher-dimensional volume question of Koh and Kovalev.
ER -