AI 中文总结
研究闭曲面上球面正交环模式的刚性问题，通过修正Nie的组合全测地曲率并利用变分原理，证明了满足特定条件的闭曲面上球面正交环模式的刚性。
AI 中文摘要
正交环模式是圆模式的自然推广。Bobenko-Hoffmann-Rörig和Bobenko为欧几里得、双曲和球面正交环模式建立了经典组合曲率的变分原理。他们的工作暗示了闭曲面上欧几里得和双曲正交环模式的刚性，而闭曲面上球面正交环模式的刚性未知。本文研究满足一定必要条件的具有胞腔分解的闭曲面上的球面正交环模式。通过修正Nie引入的组合全测地曲率，利用变分原理证明了闭曲面上球面正交环模式的刚性。
英文摘要
Orthogonal ring patterns are natural generalizations of circle patterns. Bobenko-Hoffmann-Rörig and Bobenko established the variational principles of the classical combinatorial curvature for the Euclidean, hyperbolic and spherical orthogonal ring patterns. Bobenko-Hoffmann-Rörig's work and Bobenko's work imply the rigidity of Euclidean and hyperbolic orthogonal ring patterns on closed surfaces, while the rigidity of spherical orthogonal ring patterns on closed surfaces is not known. In this paper, we study the spherical orthogonal ring patterns on closed surfaces with cellular decompositions satisfying certain necessary conditions. Using a modification of the combinatorial total geodesic curvature introduced by Nie in \cite{Nie}, we prove the rigidity of spherical orthogonal ring patterns on closed surfaces by variational principles.
