Year: 2026
Type: preprint
Abstract: Kerckhoff proved that the geodesic length function $\ell_Ω$ 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.
Source: arXiv (Cornell University)
Authors: Behera, Subash Chandra, Parsad, Shiv
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G6670785754, G7539434118
OpenAlex ID: W7168267158