Explicit solutions of Ricci flow equation for locally homogeneous $\mathbb S^1$-triples on compact Riemann surfaces

Let $(M,g)$ be a compact Riemann surfaces, and $\pi:P\to M$ be a principal $\mathbb S^1$-bundle over $M$ endowed with a connection $A$. Fixing an inner product on the Lie algebra of $\mathbb{S}^1$, the connection $A$ and metric $g$ define a Riemannian metric $g_A$ on $P$. In this article, we show that the Ricci flow equation of metric $g_A$ is equivalent to a system of differential equations. We will give an explicit solution of normalized Ricci flow equation of metric $g_A$ in the case where the base manifold is of constant curvature and the the initial connection $A$ is Yang-Mills. Finally, we will describe some asymptotic behaviors of these flows.