2005/12/17 by Sasha Anan’in, Anan'in, Sasha, Nikolay Gusevskii +1
Mathematics · #51M10 #57M50 (Secondary) #57S30 (Primary) 30F35 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.math/0512406
openalex publication_date 2005/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This article is based on the methods developed in [AGG]. We construct a complex hyperbolic structure on a trivial disc bundle over a closed orientable surface Σ (of genus 2) thus solving a long standing problem in complex hyperbolic geometry (see [Gol1, p. 583] and [Sch, p. 14]). This example answers also [Eli, Open Question 8.1] if a trivial circle bundle over a closed surface of genus >1 admits a holomorphically fillable contact structure. The constructed example M satisfies the relation 2(χ+e)=3τ which is necessary for the existence of a holomorphic section of the bundle, where χ=χΣ stands for the Euler characteristic of Σ, e=eM, for the Euler number of the bundle, and τ, for the Toledo invariant. (The relation is also valid for the series of examples constructed in [AGG].) Open question: Does there exist a holomorphic section of the bundle M?