The Moment-Weight Inequality and the Hilbert–Mumford Criterion

This book provides an introduction to geometric invariant theory from a differential geometric viewpoint.  It is inspired by certain infinite-dimensional analogues of geometric invariant theory that arise naturally in several different areas of geometry. The 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. The exposition is essentia