This chapter introduces the Kempf-Ness function and examines its negative gradient flow.

This chapter introduces the μ-stability conditions for elements of X and characterizes them in terms of the properties of the Kempf–Ness function Φx. This is the content of the generalized Kempf–Ness Theorem 7.3.

We give a condition ensuring that the operators in a nilpotent Lie algebra of linear operators on a finite dimensional vector space have a common eigenvector.

This chapter examines linear group actions on projective space, which is the classical setting of geometric invariant theory.

This chapter is devoted to the Kempf Existence and Uniqueness Theorems, which assert that the Kempf-Ness function associated to an unstable element attains its steepest negative slope along a unique geodesic ray.

This chapter introduces Mumford’s numerical weights and examines their basic properties.

This chapter introduces the moment map of a Hamiltonian group action.

This chapter examines the negative gradient flow of the square of the moment map.

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.

This chapter examines Hamiltonian torus actions and shows that in this case the Mumford weight depends continuously on the geodesic ray.