2024/09/16 by Hideki Miyachi, Miyachi, Hideki, Ken’ichi Ohshika +3
Mathematics · #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2409.10082
openalex publication_date 2024/09/16 · openalex created_date 2024/09/20 · openalex updated_date 2026/07/28
In his paper Minimal stretch maps between hyperbolic surfaces, William Thurston defined a norm on the tangent space to Teichmüller space of a hyperbolic surface, which he called the earthquake norm. This norm is obtained by assigning a length to a tangent vector after such a vector is considered as an infinitesimal earthquake deformation of the surface. This induces a Finsler metric on the Teichmüller space, called the earthquake metric. This theory was recently investigated by Huang, Ohshika, Pan and Papadopoulos. In the present paper, we study this metric from the conformal viewpoint and we adapt Thurston's theory to the case of Riemann surfaces of arbitrary genus with marked points. A complex version of the Legendre transform defined for Finsler manifolds gives an analogue of the Wolpert duality for the Weil-Petersson symplectic form, which establishes a complete analogue of Thurston's theory of the earthquake norm in the conformal setting. This paper will appear in the Annales de l'Institut Fourier