The generalized Ricci flow for three-dimensional manifolds with one Killing vector. (English) Zbl 1111.53051
Summary: We consider three-dimensional (3D) flow equations inspired by the renormalization group (RG) equations of string theory with a three-dimensional target space. By modifying the flow equations to include a U(1) gauge field, and adding carefully chosen De Turck terms, we are able to extend recent two-dimensional results of Bakas to the case of a 3D Riemannian metric with one Killing vector. In particular, we show that the RG flow with De Turck terms can be reduced to two equations: the continual Toda flow solved by Bakas, plus its linearizaton. We find exact solutions which flow to homogeneous but not always isotropic geometries.
MSC:
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 83E30 | String and superstring theories in gravitational theory |
References:
| [1] | DOI: 10.1090/conm/071/954419 · doi:10.1090/conm/071/954419 |
| [2] | DOI: 10.1090/conm/071/954419 · doi:10.1090/conm/071/954419 |
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| [9] | DOI: 10.1090/S0273-0979-99-00773-9 · Zbl 0926.53016 · doi:10.1090/S0273-0979-99-00773-9 |
| [10] | DOI: 10.1090/S0273-0979-99-00773-9 · Zbl 0926.53016 · doi:10.1090/S0273-0979-99-00773-9 |
| [11] | DOI: 10.1088/0264-9381/19/23/102 · Zbl 1039.83030 · doi:10.1088/0264-9381/19/23/102 |
| [12] | DOI: 10.1088/0264-9381/19/23/102 · Zbl 1039.83030 · doi:10.1088/0264-9381/19/23/102 |
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| [14] | for a review of this and other matters concerning the Ricci flow, see R. S. Hamilton, Surveys in Differential Geometry, Vol. 2, p. 7 (International Press, Somerville, MA, 1995). |
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