On the existence of coupled extremal Kähler metrics on ruled surfaces

Abstract: 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 $Ω<em>a$ and $Ω_b$ if and only if their parameters $(a, b)$ belong to an explicitly defined open region $\mathcal{S}</em>{\mathrm{ext}}$ in the positive real quadrant. Furthermore, we show that this existence region inherently contains the diagonal segment corresponding to classical extremal Kähler metrics.