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The Ricci flow on some generalized Wallach spaces. (English) Zbl 1323.53069

Rovenski, Vladimir (ed.) et al., Geometry and its applications. Selected papers based on the presentations at the 2nd international workshop on geometry and symbolic computation, Haifa, Israel, May 15–18, 2013. Cham: Springer (ISBN 978-3-319-04674-7/hbk; 978-3-319-04675-4/ebook). Springer Proceedings in Mathematics & Statistics 72, 3-37 (2014).
Summary: We study the asymptotic behavior of the normalized Ricci flow on generalized Wallach spaces that could be considered as a special planar dynamical system. All nonsymmetric generalized Wallach spaces can be naturally parametrized by three positive numbers \(a_{1},a_{2},a_{3}\). Our interest is to determine the type of singularity of all singular points of the normalized Ricci flow on all such spaces. Our main result gives a qualitative answer for almost all points \((a_{1},a_{2},a_{3})\) in the cube \((0,1/2] {\times} (0,1/2] {\times} (0,1/2]\). We also consider in detail some important partial cases.
For the entire collection see [Zbl 1290.53002].

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53C30 Differential geometry of homogeneous manifolds
37C10 Dynamics induced by flows and semiflows
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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