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Ricci flow of homogeneous manifolds

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Abstract

We present in this paper a general approach to study the Ricci flow on homogeneous manifolds. Our main tool is a dynamical system defined on a subset \(\mathcal H _{q,n}\) of the variety of \((q+n)\)-dimensional Lie algebras, parameterizing the space of all simply connected homogeneous spaces of dimension \(n\) with a \(q\)-dimensional isotropy, which is proved to be equivalent in a precise sense to the Ricci flow. The approach is useful to better visualize the possible (nonflat) pointed limits of Ricci flow solutions, under diverse rescalings, as well as to determine the type of the possible singularities. Ancient solutions arise naturally from the qualitative analysis of the evolution equation. We develop two examples in detail: a \(2\)-parameter subspace of \(\mathcal H _{1,3}\) reaching most of \(3\)-dimensional geometries, and a \(2\)-parameter family in \(\mathcal H _{0,n}\) of left-invariant metrics on \(n\)-dimensional compact and non-compact semisimple Lie groups.

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Acknowledgments

The author is very grateful to Ramiro Lafuente for his guidance in making the figures in this paper, and to him, Mauro Subils and the referee for very helpful comments.

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Authors and Affiliations

  1. FaMAF and CIEM, Universidad Nacional de Córdoba, Córdoba, Argentina

    Jorge Lauret

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  1. Jorge Lauret
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Correspondence to Jorge Lauret.

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This research was partially supported by grants from CONICET (Argentina) and SeCyT (Universidad Nacional de Córdoba).

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Lauret, J. Ricci flow of homogeneous manifolds. Math. Z. 274, 373–403 (2013). https://doi.org/10.1007/s00209-012-1075-z

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