A characterization of the unit ball by a K\"ahler-Einstein potential

TL;DR

This paper characterizes the unit ball among Kähler manifolds using a specific property of the Kähler-Einstein potential, and extends the Wong-Rosay theorem to complex manifolds without boundary.

Contribution

It introduces a new characterization of the unit ball via a global potential function of the Kähler-Einstein metric with constant gradient length, and extends a classical theorem to boundaryless manifolds.

Findings

Unit ball characterized by Kähler-Einstein potential with constant gradient

Extension of Wong-Rosay theorem to boundaryless complex manifolds

Universal cover of certain Kähler manifolds is the unit ball

Abstract

We will show that a universal covering of a compact K\"ahler manifold with ample canonical bundle is the unit ball if it admits a global potential function of the K\"ahler-Einstein metric whose gradient length is a minimal constant. As an application, we will extend the Wong-Rosay theorem to a complex manifold without boundary.