Yang–Mills flow

In differential geometry, the Yang–Mills flow is a gradient flow described by the Yang–Mills equations, hence a method to describe a gradient descent of the Yang–Mills action functional. Simply put, the Yang–Mills flow is a path always going in the direction of steepest descent, similar to the path of a ball rolling down a hill. This helps to find critical points, called Yang–Mills connections or instantons, which solve the Yang–Mills equations, as well as to study their stability. Illustratively, they are the points on the hill on which the ball can rest.

The Yang–Mills flow is named after Yang Chen-Ning and Robert Mills, who formulated the underlying Yang–Mills theory in 1954, although it was first studied by Michael Atiyah and Raoul Bott in 1982. It was also studied by Simon Donaldson in the context of the Kobayashi–Hitchin correspondence (or Donaldson–Uhlenbeck–Yau theorem).