Gradient flow of Einstein-Maxwell theory and Reissner-Nordstr\"om black holes

Understand

Ricci flow is a natural gradient flow of the Einstein-Hilbert action.

• Here we consider the analog for the Einstein-Maxwell action, which gives Ricci flow with a stress tensor contribution coupled to a Yang-Mills flow for the Maxwell field.
• We argue that this flow is well-posed for static spacetimes with pure electric or magnetic potentialsand show it preserves both non-extremal and extremal black hole horizons.
• In the latter case we find the flow of the near horizon geometry decouples from that of the exterior.