Geometry of bounded domains with piecewise smooth boundary

Abstract

In this paper, we study the geometry of bounded domains with piecewise smooth boundary. Specifically, we obtain the relationship between the squeezing function corresponding to polydisk and Levi flatness on bounded convex domains. As an application, we prove that a two-dimensional bounded generic convex domain with piecewise $C^2$-smooth boundary that is not smooth and admits a finite volume quotient is biholomorphic to bidisk. Moreover, we show that any Teichmüller space ${\mathcal{T}}_g$ with $g\ge 2$ cannot be biholomorphic to a bounded domain with piecewise $C^2$-smooth boundary.