Minima of geodeics length functions for non-uniform filling

Subash Chandra Behera; Shiv Parsad

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arXiv · 2607.10344

Minima of geodeics length functions for non-uniform filling

Subash Chandra Behera·Shiv Parsad

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Abstract

Kerckhoff proved that the geodesic length function $\ell_\Omega$ of a filling $\Omega$ on $S_g$ attains a unique minimum in Teichm\"uller 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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Subash Chandra Behera, Shiv Parsad. 2026-07-11. Minima of geodeics length functions for non-uniform filling. https://arxiv.org/abs/2607.10344

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