Canonical metrics and ambiKähler structures on 4-manifolds with \(U(2)\) symmetry

Brian Weber; Keaton Naff

doi:10.56754/0719-0646.2701.135

DOI:

https://doi.org/10.56754/0719-0646.2701.135

Abstract

For \(U(2)\)-invariant 4-metrics, we show that the \(B^t\)-flat metrics are very different from the other canonical metrics (Bach-flat, Einstein, extremal Kähler, etc.) We show every \(U(2)\)-invariant metric is conformal to two separate Kähler metrics, leading to ambiKähler structures. Using this observation we find new complete extremal Kähler metrics on the total spaces of \(\mathcal{O}(-1)\) and \(\mathcal{O}(+1)\) that are conformal to the Taub-bolt metric. In addition to its usual hyperKähler structure, the Taub-NUT's conformal class contains two additional complete Kähler metrics that make up an ambi-Kähler pair, making five independent compatible complex structures for the Taub-NUT, each of which is conformally Kähler.

Keywords

Cohomogeneity-1 metric , canonical metric , Taub-NUT , Bach tensor

Mathematics Subject Classification:

53C25 , 53B20 , 53C55 , 34A12

Brian Weber Institute of Mathematical Sciences, ShanghaiTech University, 393 Middle Huaxia Road, Pudong New District, Shanghai, China. https://orcid.org/0000-0002-8530-7671
Keaton Naff Massachusetts Institute of Technology (MIT), Cambridge, Massachusetts, USA. https://orcid.org/0000-0001-7474-772X

Pages: 135–163
Date Published: 2025-04-30
Vol. 27 No. 1 (2025)

V. Apostolov, D. M. J. Calderbank, and P. Gauduchon, “The geometry of weakly self-dual Kähler surfaces,” Compositio Math., vol. 135, no. 3, pp. 279–322, 2003, doi: 10.1023/A:1022251819334.

V. Apostolov, D. M. J. Calderbank, and P. Gauduchon, “Ambitoric geometry I: Einstein metrics and extremal ambikähler structures,” J. Reine Angew. Math., vol. 721, pp. 109–147, 2016, doi: 10.1515/crelle-2014-0060.

R. Bach, “Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmung-stensorbegriffs,” Math. Z., vol. 9, no. 1-2, pp. 110–135, 1921.

L. Bérard-Bergery, “Sur de nouvelles variétés riemanniennes d’Einstein,” Inst. Élie. Cartan, vol. 6, pp. 1–60, 1982.

A. L. Besse, Einstein manifolds, ser. Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer-Verlag, Berlin, 1987, vol. 10, doi: 10.1007/978-3-540-74311-8.

O. Biquard and P. Gauduchon, “On toric Hermitian ALF gravitational instantons,” Comm. Math. Phys., vol. 399, no. 1, pp. 389–422, 2023, doi: 10.1007/s00220-022-04562-z.

J.-P. Bourguignon, “Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d’Einstein,” Invent. Math., vol. 63, no. 2, pp. 263–286, 1981, doi: 10.1007/BF01393878.

R. L. Bryant, “Bochner-Kähler metrics,” J. Amer. Math. Soc., vol. 14, no. 3, pp. 623–715, 2001, doi: 10.1090/S0894-0347-01-00366-6.

E. Calabi, “Extremal Kähler metrics,” in Seminar on Differential Geometry, ser. Ann. of Math. Stud. Princeton Univ. Press, Princeton, NJ, 1982, vol. No. 102, pp. 259–290.

X. Chen, C. Lebrun, and B. Weber, “On conformally Kähler, Einstein manifolds,” J. Amer. Math. Soc., vol. 21, no. 4, pp. 1137–1168, 2008, doi: 10.1090/S0894-0347-08-00594-8.

Y. Chen and E. Teo, “A new AF gravitational instanton,” Phys. Lett. B, vol. 703, no. 3, pp. 359–362, 2011, doi: 10.1016/j.physletb.2011.07.076.

Y. Chen and E. Teo, “Five-parameter class of solutions to the vacuum Einstein equations,” Phys. Rev. D, vol. 91, no. 12, pp. 124005, 17, 2015, doi: 10.1103/PhysRevD.91.124005.

S. A. Cherkis and A. Kapustin, “Hyper-Kähler metrics from periodic monopoles,” Phys. Rev. D (3), vol. 65, no. 8, pp. 084015, 10, 2002, doi: 10.1103/PhysRevD.65.084015.

A. Derdziński, “Classification of certain compact Riemannian manifolds with harmonic curvature and nonparallel Ricci tensor,” Math. Z., vol. 172, no. 3, pp. 273–280, 1980, doi: 10.1007/BF01215090.

A. Derdziński, “Self-dual Kähler manifolds and Einstein manifolds of dimension four,” Compositio Math., vol. 49, no. 3, pp. 405–433, 1983.

T. Eguchi, P. B. Gilkey, and A. J. Hanson, “Gravitation, gauge theories and differential geometry,” Phys. Rep., vol. 66, no. 6, pp. 213–393, 1980.

T. Eguchi and A. J. Hanson, “Asymptotically flat self-dual solutions to Euclidean gravity,” Physics letters B, vol. 74, no. 3, pp. 249–251, 1978.

G. Etesi, “The topology of asymptotically locally flat gravitational instantons,” Phys. Lett. B, vol. 641, no. 6, pp. 461–465, 2006, doi: 10.1016/j.physletb.2006.08.080.

J. Fu, S.-T. Yau, and W. Zhou, “On complete constant scalar curvature Kähler metrics with Poincaré-Mok-Yau asymptotic property,” Comm. Anal. Geom., vol. 24, no. 3, pp. 521–557, 2016, doi: 10.4310/CAG.2016.v24.n3.a4.

P. Gauduchon, “The Taub-NUT ambitoric structure,” in Geometry and physics. Vol. I. ford Univ. Press, Oxford, 2018, pp. 163–187.

G. W. Gibbons and S. W. Hawking, “Classification of gravitational instanton symmetries,” Comm. Math. Phys., vol. 66, no. 3, pp. 291–310, 1979.

G. W. Gibbons and M. J. Perry, “New gravitational instantons and their interactions,” Phys. Rev. D (3), vol. 22, no. 2, pp. 313–321, 1980, doi: 10.1103/PhysRevD.22.313.

G. W. Gibbons and C. N. Pope, “CP2 as a gravitational instanton,” Comm. Math. Phys., vol. 61, no. 3, pp. 239–248, 1978.

G. W. Gibbons and S. W. Hawking, “Gravitational multi-instantons,” Physics Letters B, vol. 78, no. 4, pp. 430–432, 1978.

M. J. Gursky and J. A. Viaclovsky, “Critical metrics on connected sums of Einstein four-manifolds,” Adv. Math., vol. 292, pp. 210–315, 2016, doi: 10.1016/j.aim.2015.11.054.

S. W. Hawking, “Gravitational instantons,” Phys. Lett. A, vol. 60, no. 2, pp. 81–83, 1977, doi: 10.1016/0375-9601(77)90386-3.

A. D. Hwang and S. R. Simanca, “Distinguished Kähler metrics on Hirzebruch surfaces,” Trans. Amer. Math. Soc., vol. 347, no. 3, pp. 1013–1021, 1995, doi: 10.2307/2154885.

A. D. Hwang and S. R. Simanca, “Extremal Kähler metrics on Hirzebruch surfaces which are locally conformally equivalent to Einstein metrics,” Math. Ann., vol. 309, no. 1, pp. 97–106, 1997, doi: 10.1007/s002080050104.

C. LeBrun, “Counter-examples to the generalized positive action conjecture,” Comm. Math. Phys., vol. 118, no. 4, pp. 591–596, 1988.

C. LeBrun, “Explicit self-dual metrics on CP2#···#CP2,” J. Differential Geom., vol. 34, no. 1, pp. 223–253, 1991.

C. LeBrun, “Bach-flat Kähler surfaces,” J. Geom. Anal., vol. 30, no. 3, pp. 2491–2514, 2020, doi: 10.1007/s12220-017-9925-x.

C. W. Misner, “The flatter regions of Newman, Unti, and Tamburino’s generalized Schwarzschild space,” J. Mathematical Phys., vol. 4, pp. 924–937, 1963, doi: 10.1063/1.1704019.

J. Morrow and K. Kodaira, Complex manifolds. York-Montreal, Que.-London, 1971. Holt, Rinehart and Winston, Inc., New

N. Otoba, “Constant scalar curvature metrics on Hirzebruch surfaces,” Ann. Global Anal. Geom., vol. 46, no. 3, pp. 197–223, 2014.

D. Page, “A compact rotating gravitational instanton,” Physics Letters B, vol. 79, no. 3, pp. 235–238, 1978.

D. N. Page, “Taub-NUT instanton with an horizon,” Physics Letters B, vol. 78, no. 2-3, pp. 249–251, 1978.

J. F. Plebański and M. Demiański, “Rotating, charged, and uniformly accelerating mass in general relativity,” Ann. Physics, vol. 98, no. 1, pp. 98–127, 1976, doi: 10.1016/0003-4916(76)90240-2.

K. Schwarzschild, “On the gravitational field of a mass point according to Einstein’s theory,” Gen. Relativity Gravitation, vol. 35, no. 5, pp. 951–959, 2003, doi: 10.1023/A:1022971926521.

S.-i. Tachibana and R. C. Liu, “Notes on Kählerian metrics with vanishing Bochner curvature tensor,” Kodai Math. Sem. Rep., vol. 22, pp. 313–321, 1970, doi: 10.2996/kmj/1138846167.

B. Weber, “Asymptotic geometry of toric Kähler instantons,” 2022, arXiv:2208.00997.

B. Weber, “Analytic classification of toric Kähler instanton metrics in dimension 4,” J. Geom., vol. 114, no. 3, 2023, Art. ID 28, doi: 10.1007/s00022-023-00689-z.