On the existence of coupled extremal Kähler metrics on ruled surfaces

We investigate the existence conditions for coupled extremal Kähler metrics on minimal ruled surfaces over a genus 2 Riemann surface. Using the Calabi ansatz to reduce the coupled extremal equations to ordinary differential equations, we prove that a pair of coupled extremal metrics exists for normalized Kähler classes ${\mathrm{\Omega }}_a$ and ${\mathrm{\Omega }}_b$ if and only if their parameters $(a,b)$ belong to an explicitly defined open region ${\mathcal{S}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$ in the positive real quadrant. Furthermore, we show that this existence region inherently contains the diagonal segment corresponding to classical extremal Kähler metrics.