# Four-dimensional Einstein metrics from biconformal deformations

Archivum Mathematicum (2021)

- Volume: 057, Issue: 5, page 255-283
- ISSN: 0044-8753

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topBaird, Paul, and Ventura, Jade. "Four-dimensional Einstein metrics from biconformal deformations." Archivum Mathematicum 057.5 (2021): 255-283. <http://eudml.org/doc/297881>.

@article{Baird2021,

abstract = {Biconformal deformations take place in the presence of a conformal foliation, deforming by different factors tangent to and orthogonal to the foliation. Four-manifolds endowed with a conformal foliation by surfaces present a natural context to put into effect this process. We develop the tools to calculate the transformation of the Ricci curvature under such deformations and apply our method to construct Einstein $4$-manifolds. Examples of one particular family have ends which collapse asymptotically to $\mathbb \{R\}^2$.},

author = {Baird, Paul, Ventura, Jade},

journal = {Archivum Mathematicum},

keywords = {Einstein manifold; conformal foliation; semi-conformal map; biconformal deformation},

language = {eng},

number = {5},

pages = {255-283},

publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},

title = {Four-dimensional Einstein metrics from biconformal deformations},

url = {http://eudml.org/doc/297881},

volume = {057},

year = {2021},

}

TY - JOUR

AU - Baird, Paul

AU - Ventura, Jade

TI - Four-dimensional Einstein metrics from biconformal deformations

JO - Archivum Mathematicum

PY - 2021

PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno

VL - 057

IS - 5

SP - 255

EP - 283

AB - Biconformal deformations take place in the presence of a conformal foliation, deforming by different factors tangent to and orthogonal to the foliation. Four-manifolds endowed with a conformal foliation by surfaces present a natural context to put into effect this process. We develop the tools to calculate the transformation of the Ricci curvature under such deformations and apply our method to construct Einstein $4$-manifolds. Examples of one particular family have ends which collapse asymptotically to $\mathbb {R}^2$.

LA - eng

KW - Einstein manifold; conformal foliation; semi-conformal map; biconformal deformation

UR - http://eudml.org/doc/297881

ER -

## References

top- Baird, P., Wood, J.C., Harmonic morphisms between Riemannian manifolds, London Math. Soc. Monographs, New Series, vol. 29, Oxford Univ. Press, 2003. (2003) MR2044031
- Besse, A., Einstein Manifolds, Springer-Verlag, 1987. (1987) Zbl0613.53001
- Danielo, L., Structures Conformes, Harmonicité et Métriques d’Einstein, Ph.D. thesis, Université de Bretagne Occidentale, 2004. (2004)
- Danielo, L., 10.5802/afst.1129, Ann. Fac. Sci. Toulouse (6) 15 (3) (2006), 553–588. (2006) MR2246414DOI10.5802/afst.1129
- Dieudonné, J., Foundations of Modern Analysis, Academic Press, 1969. (1969)
- Hebey, E., Scalar curvature type problems in Riemannian geometry, Notes of a course given at the University of Rome 3. http://www.u-cergy.fr/rech/pages/hebey/.
- Hebey, E., Introduction à l’analyse non linéaire sur les variétés, Diderot, Paris, 1997. (1997)
- Hilbert, D., Die Grundlagen der Physik, Nachr. Ges. Wiss. Göttingen (1915), 395–407. (1915)
- Hitchin, N.J., 10.4310/jdg/1214432419, J. Differential Geom. 9 (1974), 435–442. (1974) DOI10.4310/jdg/1214432419
- Schoen, R., 10.4310/jdg/1214439291, J. Differential Geom. 20 (1984), 479–495. (1984) Zbl0576.53028DOI10.4310/jdg/1214439291
- Spivak, M., A Comprehensive Introduction to Riemannian Geometry, 2nd ed., Publish or Perish, Wilmongton DE, 1979. (1979)
- Thorpe, J.A., Some remarks on the Gauss-Bonnet formula, J. Math. Mech. 18 (1969), 779–786. (1969)
- Vaisman, I., 10.2996/kmj/1138035963, Kodai Math. J. 2 (1979), 26–37. (1979) DOI10.2996/kmj/1138035963
- Yamabe, H., On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 21–37. (1960) Zbl0096.37201

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