Dirichlet symbols and the nonlinear wave equation
Abstract
We develop the geometry of conformal mappings through Dirichlet operator symbols and the nonlinear wave equation. These symbols, introduced in the work of Hedenmalm and Shimorin (2020), are holomorphic functions on the bidisk associated with contractions on L2(𝔻). For Grunsky operators, the nonlinear wave equation and the related Riccati equation induce a hierarchy of of differential relations along the diagonal. We use this jet hierachy to obtain alternative coordinates and a characterization of the manifold Σ/ℂ of normalized exterior univalent functions modulo translations, without an explicit univalence test. This manifold may be viewed as an extension of the universal Teichmüller space beyond the quasicircle boundary setting. The diagonal jets also give rise to a hierarchy of \emph{diagonal series Schwarzian derivatives}. Already the fourth Schwarzian S4(ψ) leads to differential equations connected with the Weierstrass elliptic ℘ function and Painlevé I, revealing a comparison between the geometry of conformal mappings and that of complex tori. As a quantitative application of the framework, we interpret asymptotic variance through the Littlewood-Paley formula as an asymptotic density of quadratic nonlinearity energy under logarithmic averaging. We introduce Schwarzian asymptotic variance and obtain the upper bound 12.566563… for its averaged squared hyperbolically normalized amplitude, in comparison with the classical pointwise bound $36$.
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BibTeX
@misc{debnath2026dirichlet,
title = {{Dirichlet symbols and the nonlinear wave equation}},
author = {Ramlal Debnath and Haakan Hedenmalm},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2608.04666},
eprint = {2608.04666},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.04666},
}
RIS
TY - UNPB
TI - Dirichlet symbols and the nonlinear wave equation
AU - Ramlal Debnath
AU - Haakan Hedenmalm
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2608.04666
UR - https://arxiv.org/abs/2608.04666
AB - We develop the geometry of conformal mappings through Dirichlet operator symbols and the nonlinear wave equation. These symbols, introduced in the work of Hedenmalm and Shimorin (2020), are holomorphic functions on the bidisk associated with contractions on L^2(𝔻). For Grunsky operators, the nonlinear wave equation and the related Riccati equation induce a hierarchy of of differential relations along the diagonal. We use this jet hierachy to obtain alternative coordinates and a characterization of the manifold Σ/ℂ of normalized exterior univalent functions modulo translations, without an explicit univalence test. This manifold may be viewed as an extension of the universal Teichmüller space beyond the quasicircle boundary setting. The diagonal jets also give rise to a hierarchy of \emph{diagonal series Schwarzian derivatives}. Already the fourth Schwarzian S_4(ψ) leads to differential equations connected with the Weierstrass elliptic ℘ function and Painlevé I, revealing a comparison between the geometry of conformal mappings and that of complex tori. As a quantitative application of the framework, we interpret asymptotic variance through the Littlewood-Paley formula as an asymptotic density of quadratic nonlinearity energy under logarithmic averaging. We introduce Schwarzian asymptotic variance and obtain the upper bound 12.566563… for its averaged squared hyperbolically normalized amplitude, in comparison with the classical pointwise bound $36$.
ER -