AI 中文总结
本文引入推广带冠双曲曲面的翻转子曲面，发展其相关理论，建立几何递归公式，推导体积的积分与显式表示，分析其算术结构并证明相关恒等式。
AI 中文摘要
我们引入一类自然的双曲曲面，称为翻转子曲面，它推广了带冠双曲曲面（即开弦的世界面）。我们发展了它们的Teichmüller理论和模空间理论，构造了广义Weil-Petersson体积形式和Chekhov作用量，并证明所得的广义Mirzakhani体积是有限的。我们建立了三个几何递归公式——颈部截断、圆盘切除和冠提取，这些公式将这些体积用拓扑更简单的曲面的体积来表示。对于基本的多边形和环形情形，我们推导了涉及锥形勒让德函数的积分表示，以及用椭圆积分和多重对数表示的显式公式。我们进一步描述了这些体积的泰勒系数的算术结构，证明合适的特化是Kontsevich-Zagier周期，并证明了虚边界长度为2π√(-1)时的恒等式，这些恒等式推广了Do-Norbury对弦方程和伸缩子型方程的改写。
英文摘要
We introduce a natural class of hyperbolic surfaces called flippered surfaces that generalize crowned hyperbolic surfaces (i.e.: worldsheets for open strings). We develop their Teichmüller and moduli-space theory, construct generalized Weil-Petersson volume forms and Chekhov's action, and prove that the resulting generalized Mirzakhani volumes are finite. We establish three geometric recursion formulae-neck chopping, disk excision, and crown extraction-which express these volumes in terms of those of topologically simpler surfaces. For the fundamental polygonal and annular cases, we derive integral representations involving conical Legendre functions, as well as explicit formulae in terms of elliptic integrals and polylogarithms. We further describe the arithmetic structure of the Taylor coefficients of these volumes, show that suitable specializations are Kontsevich-Zagier periods, and prove identities at the imaginary boundary length $2π\sqrt{-1}$ that generalize the Do-Norbury paraphrasing of string and dilaton-type equations.
Comments80 pages, 25 figures
