  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. The Schwarzschild black
hole is an unstable fixed point of Ricci flow for static spacetimes. Here we
consider flows of the Reissner-Nordstr\"om (RN) fixed point. The magnetic RN
solution becomes a stable fixed point of the flow for sufficient charge.
However we find that the electric RN black hole is always unstable. Numerically
solving the flow starting with a spherically symmetric perturbation of a
non-extremal RN solution, we find similar behaviour in the electric case to the
Ricci flows of perturbed Schwarzschild, namely the horizon shrinks to a
singularity in finite time or expands forever. In the magnetic case, a
perturbed unstable RN solution has a similar expanding behaviour, but a
perturbation that decreases the horizon size flows to a stable black hole
solution rather than a singularity. For extremal RN we solve the near horizon
flow for spherical symmetry exactly, and see in the electric case two unstable
directions which flow to singularities in finite flow time. However, even
turning these off, and fixing the near horizon geometry to be that of RN, we
numerically show that the flows appear to become singular in the vicinity of
its horizon.
