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arXiv · 2609.18418

On the existence of coupled extremal Kähler metrics on ruled surfaces

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

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Ramesh Mete. 2026-09-16. On the existence of coupled extremal Kähler metrics on ruled surfaces. https://arxiv.org/abs/2609.18418

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