Yang-Mills flow, Ricci flow and the holonomy

Question

Is the holonomy group (based at some point) preserved along the Yang-Mills flow/ Ricci flow?

(1) For Yang-Mills case, we know that the centralizer of the holonomy $H_x$ is the isotropy group of the connection $\Gamma_A$. Hence, by the uniqueness of the flow, $\Gamma_A$ can not get smaller along the flow. But can it get larger?

(2) Can we derive anything for $H_x$ given some results of $\Gamma_A$?

(3) What about the Ricci flow case? I am not very sure about this. It seems that special holonomy means (almost) Einstein, except being Kahler. In either case, they are preserved?

Answer 1

If your compact manifold (assume irreducible WLOG, i.e. not a product) has Einstein metric, then Ricci flow just scales, so we're good. But as a counterexample in general, consider $SO(n)$ with an almost-flat torus (i.e. perturb the flat metric). The holonomy is $SO(n)$, but Ricci flow can make it flat in the limit, so that holonomy drops to trivial.
Now for the Kahler case $U(n/2)$, Ricci flow preserves Kahlerness of the metric, so that the holonomy cannot jump above $U(n/2)$. But it could possibly drop as in the above counterexample.

(I remember learning about this from Robert Bryant)