The Taub-NUT Metric Is Not Projectively Induced
Abstract
LeBrun's Kähler realization gm of the Taub--NUT metric on ℂ2 is complete, Ricci-flat and not flat. Loi, Zedda and Zuddas proved that no multiple αgm admits a Kähler immersion into a finite- or infinite-dimensional complex projective space when m>α/2, and conjectured that the same holds for every m>0. We prove the conjecture. The restriction of the Kähler potential to the axis z2=0 is governed by the Lambert W function, so exp(αΦm) has a finite radius of convergence as a power series in |z1|2 although it is real analytic on the whole half-line; the Vivanti--Pringsheim theorem forbids nonnegative Taylor coefficients, and Calabi's criterion fails. We state the mechanism, which Arezzo, Loi, Placini and Zedda recently used for radial metrics, as a general obstruction to Kähler immersions. In statistical terms the axis restriction of gm would be a natural exponential family with mean domain (0,∞) and variance function μ/(1+2mμ); the argument gives an elementary proof of the known fact, due to Bar-Lev, Bshouty and Enis, that no such family exists with variance function μ/(1+cμ) for any c>0. The result confirms onemore case of the conjecture of Loi, Salis and Zuddas that Ricci-flat projectively induced Kähler metrics are flat. The analytic core of the proof has been machine-checked in Lean~4.
Related
BibTeX
@misc{huang2026taubnut,
title = {{The Taub-NUT Metric Is Not Projectively Induced}},
author = {Shaosai Huang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21950},
eprint = {2609.21950},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21950},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The Taub-NUT Metric Is Not Projectively Induced
AU - Shaosai Huang
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21950
UR - https://arxiv.org/abs/2609.21950
AB - LeBrun's Kähler realization g_m of the Taub--NUT metric on ℂ^2 is complete, Ricci-flat and not flat. Loi, Zedda and Zuddas proved that no multiple αg_m admits a Kähler immersion into a finite- or infinite-dimensional complex projective space when m>α/2, and conjectured that the same holds for every m>0. We prove the conjecture. The restriction of the Kähler potential to the axis z_2=0 is governed by the Lambert W function, so exp(αΦ_m) has a finite radius of convergence as a power series in |z_1|^2 although it is real analytic on the whole half-line; the Vivanti--Pringsheim theorem forbids nonnegative Taylor coefficients, and Calabi's criterion fails. We state the mechanism, which Arezzo, Loi, Placini and Zedda recently used for radial metrics, as a general obstruction to Kähler immersions. In statistical terms the axis restriction of g_m would be a natural exponential family with mean domain (0,∞) and variance function μ/(1+2mμ); the argument gives an elementary proof of the known fact, due to Bar-Lev, Bshouty and Enis, that no such family exists with variance function μ/(1+cμ) for any c>0. The result confirms onemore case of the conjecture of Loi, Salis and Zuddas that Ricci-flat projectively induced Kähler metrics are flat. The analytic core of the proof has been machine-checked in Lean~4.
ER -