The moment-weight inequality and the Hilbert-Mumford criterion
Valentina Georgoulas, Joel W. Robbin, Dietmar A. Salamon
TL;DR
This paper provides a self-contained differential geometric exposition of geometric invariant theory, focusing on the moment-weight inequality, the gradient flow of the moment map squared, and the Kempf-Ness function, with minimal external references.
Contribution
It offers a comprehensive differential geometric perspective on geometric invariant theory, emphasizing key inequalities and flow methods, with minimal reliance on external results.
Findings
Establishes the moment-weight inequality relating Mumford invariants to the moment map norm
Analyzes the negative gradient flow of the squared moment map
Clarifies the role of the Kempf-Ness function in geometric invariant theory
Abstract
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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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Matrix Theory and Algorithms