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Cohomogeneity one Einstein metrics on complex projective spaces
Abstract
We study Einstein metrics on complex projective spaces that are invariant under cohomogeneity one actions of compact connected Lie groups, under the assumption that the singular orbits are totally geodesic. These actions were classified by Takagi into five models. For each of them, we write the Einstein equation for diagonal invariant metrics and determine the corresponding smoothness conditions at the singular orbits. Our main result is the nonexistence of smooth globally defined diagonal invariant Einstein metrics with totally geodesic singular orbits in four of the five models and a necessary condition for global existence in the remaining one.
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BibTeX
@misc{araujo2026cohomogeneity,
title = {{Cohomogeneity one Einstein metrics on complex projective spaces}},
author = {Anderson L. A. de Araujo and Brian Grajales and Lino Grama},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.28539},
eprint = {2605.28539},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.28539},
}
RIS
TY - UNPB
TI - Cohomogeneity one Einstein metrics on complex projective spaces
AU - Anderson L. A. de Araujo
AU - Brian Grajales
AU - Lino Grama
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2605.28539
UR - https://arxiv.org/abs/2605.28539
AB - We study Einstein metrics on complex projective spaces that are invariant under cohomogeneity one actions of compact connected Lie groups, under the assumption that the singular orbits are totally geodesic. These actions were classified by Takagi into five models. For each of them, we write the Einstein equation for diagonal invariant metrics and determine the corresponding smoothness conditions at the singular orbits. Our main result is the nonexistence of smooth globally defined diagonal invariant Einstein metrics with totally geodesic singular orbits in four of the five models and a necessary condition for global existence in the remaining one.
ER -