Complete K\"ahler-Einstein metric on Stein manifolds with negative curvature

TL;DR

This paper proves the existence and deformation of complete negative K"ahler-Einstein metrics on Stein manifolds with negatively pinched holomorphic sectional curvature, using the normalized K"ahler-Ricci flow.

Contribution

It establishes the existence of such metrics and shows they can be obtained via deformation from any initial K"ahler metric.

Findings

Existence of complete negative K"ahler-Einstein metrics on Stein manifolds.

Deformation of any K"ahler metric to the Einstein metric via K"ahler-Ricci flow.

Application of the flow to achieve the Einstein metric.

Abstract

We show the existence of complete negative K\"ahler-Einstein metric on Stein manifolds with negatively pinched holomorphic sectional curvature. We prove that any K\"ahler metrics on such manifolds can be deformed to the complete negative K\"ahler-Einstein metric using the normalized K\"ahler-Ricci flow.