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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 Ωa and Ωb if and only if their parameters $(a, b)$ belong to an explicitly defined open region 𝒮ext 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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BibTeX
@misc{mete2026existence,
title = {{On the existence of coupled extremal Kähler metrics on ruled surfaces}},
author = {Ramesh Mete},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18418},
eprint = {2609.18418},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18418},
}
RIS
TY - UNPB
TI - On the existence of coupled extremal Kähler metrics on ruled surfaces
AU - Ramesh Mete
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18418
UR - https://arxiv.org/abs/2609.18418
AB - 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 𝒮_ext in the positive real quadrant. Furthermore, we show that this existence region inherently contains the diagonal segment corresponding to classical extremal Kähler metrics.
ER -