  We present some results that complement our prequels
[arXiv:1809.08425,arXiv:1907.05770] on holomorphic vector bundles. We apply the
method of the Quot-scheme limit of Fubini-Study metrics developed therein to
provide a generalisation to the singular case of the result originally obtained
by X.W. Wang for the smooth case, which states that the existence of balanced
metrics is equivalent to the Gieseker stability of the vector bundle. We also
prove that the Bergman 1-parameter subgroups form subgeodesics in the space of
hermitian metrics. This paper also contains a review of techniques developed in
[arXiv:1809.08425,arXiv:1907.05770] and how they correspond to their
counterparts developed in the study of the Yau-Tian-Donaldson conjecture.
