AI 中文总结
研究非均匀填充的测地线长度函数最小值，针对4正则拓扑均匀填充提出基本优化方法，还利用胖图和优化技术分析两类非均匀4正则填充，明确计算最小值并证明在三角形曲面取得。
AI 中文摘要
克尔霍夫证明了在Teichmüller空间中，$S_g$上填充$\Omega$的测地线长度函数$\ell_\Omega$有唯一最小值。埃内斯托·吉隆多等人利用儿童画和格罗滕迪克 - 贝利伊曲面的代数机制计算了均匀填充的这些最小值。我们提出一种针对4正则拓扑均匀填充的基本优化方法，绕过该框架。此外，我们使用胖图和优化技术分析两类非均匀4正则填充。明确计算其最小值，并证明在这两类中，这些长度函数的最小值在三角形曲面上取得。
英文摘要
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.
Comments15 pages, 6 figures
