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Abiev, N. A.; Arvanitoyeorgos, A.; Nikonorov, Yu. G.; Siasos, P.

The dynamics of the Ricci flow on generalized Wallach spaces. (English) Zbl 1327.53062

Differ. Geom. Appl. 35, Suppl., 26-43 (2014).
Summary: We consider the asymptotic behavior of the normalized Ricci flow on generalized Wallach spaces that could be considered as a special planar dynamical system. All non-symmetric generalized Wallach spaces can be naturally parameterized 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 \((0,1/2]\times(0,1/2]\times(0,1/2]\).

Cited in 8 Documents

MSC:

53C30 Differential geometry of homogeneous manifolds
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
37C10 Dynamics induced by flows and semiflows
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
14P05 Real algebraic sets

Keywords:

generalized Wallach space; Einstein metric; Ricci flow; planar dynamical system; singular point; real algebraic surface
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\textit{N. A. Abiev} et al., Differ. Geom. Appl. 35, 26--43 (2014; Zbl 1327.53062)
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