The purpose of this paper is to investigate canonical metrics on a semi-stable vector bundle $E$ over a compact Kähler manifold $X$. It is shown that if $E$ is semi-stable, then Donaldson’s functional is bounded from below. This implies that $E$ admits an approximate Hermitian-Einstein structure, generalizing a classic result of Kobayashi for projective manifolds to the Kähler case. As an application some basic properties of semi-stable vector bundles over compact Kähler manifolds are established, such as the fact that semi-stability is preserved under certain exterior and symmetric products.
"Existence of approximate Hermitian-Einstein structures on semi-stable bundles." Asian J. Math. 18 (5) 859 - 884, November 2014.