Twisted and coupled constant scalar curvature Kähler metrics on minimal ruled surfaces

In this paper, we study the existence of twisted constant scalar curvature Kähler (cscK) metrics and non-existence of coupled cscK metrics on minimal ruled surfaces over a Riemann surface of genus $2$. Moreover, we give a bound for the Chen-Cheng invariant related to Chen’s continuity path for cscK problem on these ruled surfaces.