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Ambitoric geometry I: Einstein metrics and extremal ambikaehler structures
- Source :
- Apostolov, V, Calderbank, D & Gauduchon, P 2016, ' Ambitoric geometry I : Einstein metrics and extremal ambikahler structures ', Journal Fur Die Reine Und Angewandte Mathematik, vol. 2016, no. 721, pp. 109-147 . https://doi.org/10.1515/crelle-2014-0060
- Publication Year :
- 2013
- Publisher :
- HAL CCSD, 2013.
-
Abstract
- We present a local classification of conformally equivalent but oppositely oriented 4-dimensional Kaehler metrics which are toric with respect to a common 2-torus action. In the generic case, these "ambitoric" structures have an intriguing local geometry depending on a quadratic polynomial q and arbitrary functions A and B of one variable. We use this description to classify Einstein 4-metrics which are hermitian with respect to both orientations, as well a class of solutions to the Einstein-Maxwell equations including riemannian analogues of the Plebanski-Demianski metrics. Our classification can be viewed as a riemannian analogue of a result in relativity due to R. Debever, N. Kamran, and R. McLenaghan, and is a natural extension of the classification of selfdual Einstein hermitian 4-manifolds, obtained independently by R. Bryant and the first and third authors. These Einstein metrics are precisely the ambitoric structures with vanishing Bach tensor, and thus have the property that the associated toric Kaehler metrics are extremal (in the sense of E. Calabi). Our main results also classify the latter, providing new examples of explicit extremal Kaehler metrics. For both the Einstein-Maxwell and the extremal ambitoric structures, A and B are quartic polynomials, but with different conditions on the coefficients. In the sequel to this paper we consider global examples, and use them to resolve the existence problem for extremal Kaehler metrics on toric 4-orbifolds with second betti number b2=2.<br />31 pages, 1 figure, partially replaces arXiv:1010.0992
- Subjects :
- Mathematics - Differential Geometry
General Mathematics
[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph]
FOS: Physical sciences
General Relativity and Quantum Cosmology (gr-qc)
01 natural sciences
General Relativity and Quantum Cosmology
symbols.namesake
Theoretical physics
[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]
0103 physical sciences
FOS: Mathematics
53C55, 53C25, 53B35, 32Q15
0101 mathematics
Einstein
[MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph]
Mathematics::Symplectic Geometry
Mathematical Physics
Mathematics
[PHYS.GRQC] Physics [physics]/General Relativity and Quantum Cosmology [gr-qc]
010308 nuclear & particles physics
Applied Mathematics
010102 general mathematics
Mathematical Physics (math-ph)
[PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph]
16. Peace & justice
Differential Geometry (math.DG)
[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]
symbols
[PHYS.GRQC]Physics [physics]/General Relativity and Quantum Cosmology [gr-qc]
Mathematics::Differential Geometry
[MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG]
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Journal :
- Apostolov, V, Calderbank, D & Gauduchon, P 2016, ' Ambitoric geometry I : Einstein metrics and extremal ambikahler structures ', Journal Fur Die Reine Und Angewandte Mathematik, vol. 2016, no. 721, pp. 109-147 . https://doi.org/10.1515/crelle-2014-0060
- Accession number :
- edsair.doi.dedup.....61411e7860b1b430a2c9966c891c42d2