Affine Hermitian-Einstein Metrics

TL;DR

This paper develops a theory for stable flat vector bundles and affine Hermitian-Einstein metrics on special affine manifolds, extending classical results from Kähler and non-Kähler complex geometry.

Contribution

It introduces a new stability concept based solely on flat subbundles and proves an existence theorem for affine Hermitian-Einstein metrics analogous to Donaldson-Uhlenbeck-Yau.

Findings

Established a parallel to Donaldson-Uhlenbeck-Yau theorem for affine manifolds.

Defined a simplified stability condition involving only flat subbundles.

Proved existence of affine Hermitian-Einstein metrics under stability assumptions.

Abstract

We develop a theory of stable bundles and affine Hermitian-Einstein metrics for flat vector bundles over a special affine manifold (a manifold admitting an atlas whose gluing maps are all locally constant volume-preserving affine maps). Our paper presents a parallel to Donaldson-Uhlenbeck-Yau's proof of the existence of Hermitian-Einstein metrics on K\"ahler manifolds, and the extension of this theorem by Li-Yau to the non-K\"ahler complex case of Gauduchon metrics. Our definition of stability involves only flat vector subbundles (and not singular subsheaves), and so is simpler than the complex case in some places.