Ricci flow of homogeneous manifolds

Details Ricci flow of homogeneous manifolds Ricci flow of homogeneous manifolds

2011-12-26

arxiv.org/abs/1112.5900v2

Mathematics

Differential Geometry

30 pages, 11 figures; new version, many references added, section on Ricci solitons removed (to be included in a forthcoming p

Scientific paper

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 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 H(1,3) reaching many 3-dimensional geometries, and a 2-parameter family in H(0,n) of left-invariant metrics on n-dimensional compact and non-compact semisimple Lie groups.

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