Ricci Flow of 3-D Manifolds with One Killing Vector

Details Ricci Flow of 3-D Manifolds with One Killing Vector Ricci Flow of 3-D Manifolds with One Killing Vector

2004-09-28

arxiv.org/abs/hep-th/0409293v1

Physics

High Energy Physics

High Energy Physics - Theory

Scientific paper

We implement a suggestion by Bakas and consider the Ricci flow of 3-d manifolds with one Killing vector by dimensional reduction to the corresponding flow of a 2-d manifold plus scalar (dilaton) field. By suitably modifying the flow equations in order to make them manifestly parabolic, we are able to show that the equations for the 2-d geometry can be put in the form explicitly solved by Bakas using a continual analogue of the Toda field equations. The only remaining equation, namely that of the scale factor of the extra dimension, is a linear equation that can be readily solved using standard techniques once the 2-geometry is specified. We illustrate the method with a couple of specific examples.

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