Gradient flow and Bogomolny bounds for quantum metric actions

We formulate gradient flow dynamics generated by two natural actions of the quantum metric for an isolated set of Bloch bands. Specializing to two spatial dimensions, we derive Bogomolny-type lower bounds that relate these actions to the Chern number and show that the bounds are saturated by (anti-)holomorphic projector configurations. Along the flows, the actions decrease monotonically while the Chern number is conserved, giving a constructive route to simplify models within a fixed topological phase toward canonical, low-complexity representatives.