The moment-weight inequality and the Hilbert-Mumford criterion
This paper gives an essentially self-contained exposition (except for an appeal to the Lojasiewicz gradient inequality) of geometric invariant theory from a differential geometric viewpoint. Central ingredients are the moment-weight inequality (relating the Mumford numerical invariants to the norm of the moment map), the negative gradient flow of the moment map squared, and the Kempf-Ness function.
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