Ricci flow of homogeneous manifolds
By Jorge Lauret. 2011
68 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/faeb1455b722ae6819fa89a654aa3cf8b2aab6f3
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Astronomy and Astrophysics
In Geometry and Topology
Citations in the corpus
- The long‐time behavior of the homogeneous pluriclosed flow (2017, background): “Following the ideas in [19], we explain how the normalized solutions can be regarded as solutions to a similar equation:”