Abstract
Cotton flow tends to evolve a given initial metric on a three manifold to a conformally flat one. Here we expound upon the earlier work on Cotton flow and study the linearized version of it around a generic initial metric by employing a modified form of the DeTurck trick. We show that the flow around the flat space, as a critical point, reduces to an anisotropic generalization of linearized KdV equation with complex dispersion relations one of which is an unstable mode, rendering the flat space unstable under small perturbations. We also show that Einstein spaces and some conformally flat non-Einstein spaces are linearly unstable. We refine the gradient flow formalism and compute the second variation of the entropy and show that generic critical points are extended Cotton solitons. We study some properties of these solutions and find a Topologically Massive soliton that is built from Cotton and Ricci solitons. In the Lorentzian signature, we also show that the pp-wave metrics are both Cotton and Ricci solitons.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960) 21.
R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982) 255.
R.S. Hamilton, The Ricci flow on surfaces, Contemp. Math. 71 (1988) 237.
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, math.DG/0211159 [INSPIRE].
G. Perelman, Ricci flow with surgery on three-manifolds, math.DG/0303109 [INSPIRE].
W.P. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc. 6 (1982) 357 [INSPIRE].
D. Friedan, Nonlinear models in 2 + ε dimensions, Phys. Rev. Lett. 45 (1980) 1057 [INSPIRE].
M. Headrick and T. Wiseman, Ricci flow and black holes, Class. Quant. Grav. 23 (2006) 6683 [hep-th/0606086] [INSPIRE].
N. Lashkari and A. Maloney, Topologically massive gravity and Ricci-Cotton flow, Class. Quant. Grav. 28 (2011) 105007 [arXiv:1007.1661] [INSPIRE].
E. Woolgar, Some applications of Ricci flow in physics, Can. J. Phys. 86 (2008) 645 [arXiv:0708.2144] [INSPIRE].
J. Gegenberg and G. Kunstatter, Using 3D stringy gravity to understand the Thurston conjecture, Class. Quant. Grav. 21 (2004) 1197 [hep-th/0306279] [INSPIRE].
S. Das and S. Kar, Bach flows of product manifolds, Int. J. Geom. Meth. Mod. Phys. 09 (2012) 1250039 [arXiv:1012.4244] [INSPIRE].
A.U.O. Kisisel, O. Sarioglu and B. Tekin, Cotton flow, Class. Quant. Grav. 25 (2008) 165019 [arXiv:0803.1603] [INSPIRE].
J. Isenberg and M. Jackson, Ricci flow on locally homogeneous geometries on closed manifolds, J. Diff. Geom. 35 (1992) 723.
D.M. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Diff. Geom. 18 (1983) 157.
I. Bakas, F. Bourliot, D. Lüst and M. Petropoulos, Geometric flows in Hořava-Lifshitz gravity, JHEP 04 (2010) 131 [arXiv:1002.0062] [INSPIRE].
I. Bakas, F. Bourliot, D. Lüst and M. Petropoulos, Mixmaster universe in Hořava-Lifshitz gravity, Class. Quant. Grav. 27 (2010) 045013 [arXiv:0911.2665] [INSPIRE].
R. Cartas-Fuentevilla, On the gravitational Chern-Simons action as entropy functional for three-manifolds and the demystification of the Hořava-Lifshitz gravity, arXiv:1304.6678 [INSPIRE].
E. Calviño-Louzao, E. Garcia-Rio and R. Vázquez-Lorenzo, A note on compact Cotton solitons, Class. Quant. Grav. 29 (2012) 205014 [INSPIRE].
E. Calviño-Louzao, L.M. Hervella, J. Seoane-Bascoy and R. Vázquez-Lorenzo, Homogeneous Cotton solitons, J. Phys. A 46 (2013) 285204 [arXiv:1303.3872].
E. Garcia-Rio, P.B. Gilkey and S. Nikcevic, Homogeneity of Lorentzian three-manifolds with recurrent curvature, Math. Nachr. 287 (2014) 32 [arXiv:1210.7764].
S. Deser and B. Tekin, Energy in topologically massive gravity, Class. Quant. Grav. 20 (2003) L259 [gr-qc/0307073] [INSPIRE].
H. Cebeci, O. Sarioglu and B. Tekin, Negative mass solitons in gravity, Phys. Rev. D 73 (2006) 064020 [hep-th/0602117] [INSPIRE].
F.R. Gantmacher, The theory of matrices, Chelsea Publishing Company, New York U.S.A. (1959).
S. Deser, R. Jackiw and S. Templeton, Topologically massive gauge theories, Annals Phys. 140 (1982) 372 [INSPIRE].
S. Deser, R. Jackiw and S. Templeton, Three-dimensional massive gauge theories, Phys. Rev. Lett. 48 (1982) 975 [INSPIRE].
M. Gurses, Godel type metrics in three dimensions, Gen. Rel. Grav. 42 (2010) 1413 [arXiv:0812.2576] [INSPIRE].
M. Gürses, T.C. Sisman and B. Tekin, AdS-plane wave and pp-wave solutions of generic gravity theories, Phys. Rev. D 90 (2014) 124005 [arXiv:1407.5301] [INSPIRE].
H. Afshar, B. Cvetkovic, S. Ertl, D. Grumiller and N. Johansson, Conformal Chern-Simons holography, Phys. Rev. D 85 (2012) 064033 [arXiv:1110.5644] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1502.02514
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Kilicarslan, E., Dengiz, S. & Tekin, B. More on Cotton flow. J. High Energ. Phys. 2015, 136 (2015). https://doi.org/10.1007/JHEP06(2015)136
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2015)136