Kempf–Ness theorem

In algebraic geometry, the Kempf–Ness theorem, introduced by Kempf and Ness (1979), gives a criterion for the stability of a vector in a representation of a complex reductive group. If the complex vector space is given a norm that is invariant under a maximal compact subgroup of the reductive group, then the Kempf–Ness theorem states that a vector is stable if and only if the norm attains a minimum value on the orbit of the vector.