Quasiconformal deformations of chordal Loewner chains
Abstract
We establish first-order variational formulas for the driving function and the half-plane capacity of a chordal Loewner chain under quasiconformal deformations. Under the standard normalization fixing $0,1$ and ∞, the formulas hold for every bounded Beltrami differential. If the curve has positive area, the formulas contain additional integrals over the curve; these terms vanish when the curve has zero area or when the differential is extended by zero on the curve. The proof uses the normalized Ahlfors--Bers variation together with a Stoilow factorization adapted to the slit domain. We then consider the asymptotic normalization for differentials whose symmetric extensions to ℂ have the form ν(z)=a(z̅/z)n−1+ν0(z), where a∈(−1,1), n≥1 is an integer, and ν0∈ Lr(ℂ) for 0<r<2. The asymptotically normalized map differs from the standardly normalized map by a positive dilation. Using this relation, the variational formulas in the second normalization follow directly from the standardly normalized case.
Related
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BibTeX
@misc{han2026quasiconformal,
title = {{Quasiconformal deformations of chordal Loewner chains}},
author = {Yong Han and Renxin Jiang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12021},
eprint = {2610.12021},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12021},
}
RIS
TY - UNPB
TI - Quasiconformal deformations of chordal Loewner chains
AU - Yong Han
AU - Renxin Jiang
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12021
UR - https://arxiv.org/abs/2610.12021
AB - We establish first-order variational formulas for the driving function and the half-plane capacity of a chordal Loewner chain under quasiconformal deformations. Under the standard normalization fixing $0,1$ and ∞, the formulas hold for every bounded Beltrami differential. If the curve has positive area, the formulas contain additional integrals over the curve; these terms vanish when the curve has zero area or when the differential is extended by zero on the curve. The proof uses the normalized Ahlfors--Bers variation together with a Stoilow factorization adapted to the slit domain. We then consider the asymptotic normalization for differentials whose symmetric extensions to ℂ have the form ν(z)=a(z̅/z)^n−1+ν_0(z), where a∈(−1,1), n≥1 is an integer, and ν_0∈ L^r(ℂ) for 0<r<2. The asymptotically normalized map differs from the standardly normalized map by a positive dilation. Using this relation, the variational formulas in the second normalization follow directly from the standardly normalized case.
ER -