Geodesic
Bach Flow on Homogeneous Products
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020) Cet article a éte moissonné depuis la source Math-Net.Ru

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Qualitative behavior of Bach flow is established on compact four-dimensional locally homogeneous product manifolds. This is achieved by lifting to the homogeneous universal cover and, in most cases, capitalizing on the resultant group structure. The resulting system of ordinary differential equations is carefully analyzed on a case-by-case basis, with explicit solutions found in some cases. Limiting behavior of the metric and the curvature are determined in all cases. The behavior on quotients of $\mathbb{R} \times \mathbb{S}^3$ proves to be the most challenging and interesting.
Keywords: high-order geometric flows, Bach flow, locally homogeneous manifold, three-dimensional Lie group. 
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     author = {Dylan Helliwell},
     title = {Bach {Flow} on {Homogeneous} {Products}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a26/}
}
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Dylan Helliwell. Bach Flow on Homogeneous Products. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a26/

[1] Bahuaud E., Helliwell D., “Short-time existence for some higher-order geometric flows”, Comm. Partial Differential Equations, 36 (2011), 2189–2207, arXiv: 1010.4287 | DOI | MR | Zbl

[2] Bahuaud E., Helliwell D., “Uniqueness for some higher-order geometric flows”, Bull. Lond. Math. Soc., 47 (2015), 980–995, arXiv: 1407.4406 | DOI | MR | Zbl

[3] Bour V., Fourth order curvature flows and geometric applications, arXiv: 1012.0342

[4] Cao X., Ni Y., Saloff-Coste L., “Cross curvature flow on locally homogenous three-manifolds. I”, Pacific J. Math., 236 (2008), 263–281, arXiv: 0708.1922 | DOI | MR | Zbl

[5] Cao X., Saloff-Coste L., “Backward Ricci flow on locally homogeneous 3-manifolds”, Comm. Anal. Geom., 17 (2009), 305–325, arXiv: 0810.3352 | DOI | MR | Zbl

[6] Cao X., Saloff-Coste L., “Cross curvature flow on locally homogeneous three-manifolds (II)”, Asian J. Math., 13 (2009), 421–458, arXiv: 0805.3380 | DOI | MR | Zbl

[7] Das S., Kar S., “Bach flows of product manifolds”, Int. J. Geom. Methods Mod. Phys., 9 (2012), 1250039, 18 pp., arXiv: 1012.4244 | DOI | MR | Zbl

[8] Gimre K., Guenther C., Isenberg J., “Second-order renormalization group flow of three-dimensional homogeneous geometries”, Comm. Anal. Geom., 21 (2013), 435–467, arXiv: 1205.6507 | DOI | MR | Zbl

[9] Glickenstein D., Payne T.L., “Ricci flow on three-dimensional, unimodular metric Lie algebras”, Comm. Anal. Geom., 18 (2010), 927–961, arXiv: 0909.0938 | DOI | MR | Zbl

[10] Ho P.T., “Bach flow”, J. Geom. Phys., 133 (2018), 1–9 | DOI | MR

[11] Isenberg J., Jackson M., “Ricci flow of locally homogeneous geometries on closed manifolds”, J. Differential Geom., 35 (1992), 723–741 | DOI | MR | Zbl

[12] Isenberg J., Jackson M., Lu P., “Ricci flow on locally homogeneous closed 4-manifolds”, Comm. Anal. Geom., 14 (2006), 345–386, arXiv: math.DG/0502170 | DOI | MR | Zbl

[13] Kişisel A.U.O., Sarı oğlu O., Tekin B., “Cotton flow”, Classical Quantum Gravity, 25 (2008), 165019, 15 pp., arXiv: 0803.1603 | DOI | MR

[14] Knopf D., McLeod K., “Quasi-convergence of model geometries under the Ricci flow”, Comm. Anal. Geom., 9 (2001), 879–919 | DOI | MR | Zbl

[15] Lee J.M., Riemannian manifolds. An introduction to curvature, Graduate Texts in Mathematics, 176, Springer-Verlag, New York, 1997 | DOI | MR | Zbl

[16] Lee J.M., Introduction to smooth manifolds, Graduate Texts in Mathematics, 218, 2nd ed., Springer, New York, 2013 | DOI | MR | Zbl

[17] Lopez C., “Ambient obstruction flow”, Trans. Amer. Math. Soc., 370 (2018), 4111–4145, arXiv: 1506.01979 | DOI | MR | Zbl

[18] Milnor J., “Curvatures of left invariant metrics on Lie groups”, Adv. Math., 21 (1976), 293–329 | DOI | MR | Zbl

[19] Ryan Jr. M.P., Shepley L.C., Homogeneous relativistic cosmologies, Princeton Series in Physics, Princeton University Press, Princeton, N.J., 1975 | MR

[20] Streets J.D., “The gradient flow of $\int_M|{\rm Rm}|^2$”, J. Geom. Anal., 18 (2008), 249–271 | DOI | MR | Zbl