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

Quot-scheme limit of Fubini-Study metrics and its applications to balanced metrics