PreprintCC BY 4.0arXiv
Minima of geodesic length functions for non-uniform filling systems
Abstract
Kerckhoff proved that the geodesic length function ℓΩ of a filling Ω on Sg attains a unique minimum in Teichmüller space. Recent work of Ernesto Girondo et al. computed these minima for uniform fillings using the algebraic machinery of dessins d'enfants and Grothendieck-Belyi surfaces. We present an elementary optimization approach for $4$-regular topological uniform fillings, bypassing this framework. Furthermore, we analyze two special classes of non-uniform $4$-regular fillings using fat graphs and optimization techniques. We explicitly compute their minima and prove that in both classes, the minimum of these length functions is attained at a triangle surface.
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BibTeX
@misc{behera2026minima,
title = {{Minima of geodesic length functions for non-uniform filling systems}},
author = {Subash Chandra Behera and Shiv Parsad},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.10344},
eprint = {2607.10344},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.10344},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Minima of geodesic length functions for non-uniform filling systems
AU - Subash Chandra Behera
AU - Shiv Parsad
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2607.10344
UR - https://arxiv.org/abs/2607.10344
AB - Kerckhoff proved that the geodesic length function ℓ_Ω of a filling Ω on S_g attains a unique minimum in Teichmüller space. Recent work of Ernesto Girondo et al. computed these minima for uniform fillings using the algebraic machinery of dessins d'enfants and Grothendieck-Belyi surfaces. We present an elementary optimization approach for $4$-regular topological uniform fillings, bypassing this framework. Furthermore, we analyze two special classes of non-uniform $4$-regular fillings using fat graphs and optimization techniques. We explicitly compute their minima and prove that in both classes, the minimum of these length functions is attained at a triangle surface.
ER -