Geometry of bounded generic domains with piecewise smooth boundary

TL;DR

This paper investigates the geometry of bounded domains with piecewise smooth boundaries, establishing relationships between the squeezing function, Levi flatness, and biholomorphic equivalences to bidisks and Teichmüller spaces.

Contribution

It provides new insights into the geometric properties of bounded generic convex domains and their biholomorphic classifications, especially relating to the bidisk and Teichmüller spaces.

Findings

Bounded generic convex domains with piecewise C^2 boundary and finite volume quotients are biholomorphic to bidisks.

The squeezing function relates to Levi flatness on these domains.

Teichmüller spaces with genus g ≥ 2 cannot be biholomorphic to such bounded domains.

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 generic convex domains. As an application, we prove that a two dimensional bounded generic convex domain with piecewise $C^2$-smooth boundary that admits a finite volume quotient is biholomorphic to bidisk. Moreover, we show that any Teichm$\ddot{\operatorname{u}}$ller space $\mathcal{T}_g$ with $g\geq2$ can not be biholomorphic to a bounded generic domain with piecewise $C^2$-smooth boundary.