Geodesic
Four-dimensional Einstein metrics from biconformal deformations
Archivum mathematicum, Tome 57 (2021) no. 5, pp. 255-283
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $\mathbb{R}^2$.
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 $\mathbb{R}^2$.
DOI : 10.5817/AM2021-5-255
Classification : 53C12, 53C18, 53C25
Keywords: Einstein manifold; conformal foliation; semi-conformal map; biconformal deformation 
@article{10_5817_AM2021_5_255,
     author = {Baird, Paul and Ventura, Jade},
     title = {Four-dimensional {Einstein} metrics from biconformal deformations},
     journal = {Archivum mathematicum},
     pages = {255--283},
     year = {2021},
     volume = {57},
     number = {5},
     doi = {10.5817/AM2021-5-255},
     mrnumber = {4346113},
     zbl = {07442414},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2021-5-255/}
}
TY  - JOUR
AU  - Baird, Paul
AU  - Ventura, Jade
TI  - Four-dimensional Einstein metrics from biconformal deformations
JO  - Archivum mathematicum
PY  - 2021
SP  - 255
EP  - 283
VL  - 57
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.5817/AM2021-5-255/
DO  - 10.5817/AM2021-5-255
LA  - en
ID  - 10_5817_AM2021_5_255
ER  - 
%0 Journal Article
%A Baird, Paul
%A Ventura, Jade
%T Four-dimensional Einstein metrics from biconformal deformations
%J Archivum mathematicum
%D 2021
%P 255-283
%V 57
%N 5
%U http://geodesic.mathdoc.fr/articles/10.5817/AM2021-5-255/
%R 10.5817/AM2021-5-255
%G en
%F 10_5817_AM2021_5_255
Baird, Paul; Ventura, Jade. Four-dimensional Einstein metrics from biconformal deformations. Archivum mathematicum, Tome 57 (2021) no. 5, pp. 255-283. doi: 10.5817/AM2021-5-255

[1] Baird, P., Wood, J.C.: Harmonic morphisms between Riemannian manifolds. London Math. Soc. Monographs, New Series, vol. 29, Oxford Univ. Press, 2003. | MR

[2] Besse, A.: Einstein Manifolds. Springer-Verlag, 1987. | Zbl

[3] Danielo, L.: Structures Conformes, Harmonicité et Métriques d’Einstein. Ph.D. thesis, Université de Bretagne Occidentale, 2004.

[4] Danielo, L.: Construction de métriques d’Einstein à partir de transformations biconformes. Ann. Fac. Sci. Toulouse (6) 15 (3) (2006), 553–588. | DOI | MR

[5] Dieudonné, J.: Foundations of Modern Analysis. Academic Press, 1969.

[6] 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/

[7] Hebey, E.: Introduction à l’analyse non linéaire sur les variétés. Diderot, Paris, 1997.

[8] Hilbert, D.: Die Grundlagen der Physik. Nachr. Ges. Wiss. Göttingen (1915), 395–407.

[9] Hitchin, N.J.: On compact four-dimensional Einstein manifolds. J. Differential Geom. 9 (1974), 435–442. | DOI

[10] Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differential Geom. 20 (1984), 479–495. | DOI | Zbl

[11] Spivak, M.: A Comprehensive Introduction to Riemannian Geometry. 2nd ed., Publish or Perish, Wilmongton DE, 1979.

[12] Thorpe, J.A.: Some remarks on the Gauss-Bonnet formula. J. Math. Mech. 18 (1969), 779–786.

[13] Vaisman, I.: Conformal foliations. Kodai Math. J. 2 (1979), 26–37. | DOI

[14] Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12 (1960), 21–37. | Zbl

Cité par Sources :