The Moment Weight Inequality and the Hilbert–Mumford Criterion : GIT from the Differential Geometric Viewpoint 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 essentially self contained except for an appeal to the Lojasiewicz gradient inequality. A broad variety of examples illustrate the theory and five appendices cover essential topics that go beyond the basic concepts of differential geometry. The comprehensive bibliography will be a valuable resource for researchers. The book is addressed to graduate students and researchers interested in geometric invariant theory and related subjects. It will be easily accessible to readers with a basic understanding of differential geometry and does not require any knowledge of algebraic geometry.