Flippered hyperbolic surfaces and renormalized volumes of their moduli spaces I

Abstract

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 Teichm\"uller 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 $2\pi\sqrt{-1}$ that generalize the Do-Norbury paraphrasing of string and dilaton-type equations.