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Narasiham and Seshadri, Donaldson, Uhlenbeck and Yau proved that an irreducible holomorphic vector bundle E over a Kahler manifold X is Mumford stable if and only if there exists a Hermitian Einstein metric. In order to uncover stability from geometric invariant theory point of view, we need to study Gieseker stability (Mumford stability is only a linearized version of Gieseker stability in some suitable sense.) In [Leung], the author discovered the following equation

Bando–Futaki invariants and Kähler–Einstein metric