NDLI: On the compactness of the set of invariant Einstein metrics

Content Provider	Springer Nature Link
Author	Graev, M. M.
Copyright Year	2013
Abstract	Let $$M = G/H$$ be a connected simply connected homogeneous manifold of a compact, not necessarily connected Lie group $$G$$ . We will assume that the isotropy $$H$$ -module $$\mathfrak{g/h }$$ has a simple spectrum, i.e. irreducible submodules are mutually non-equivalent. There exists a convex Newton polytope $$N=N(G,H)$$ , which was used for the estimation of the number of isolated complex solutions of the algebraic Einstein equation for invariant metrics on $$G/H$$ (up to scaling). Using the moment map, we identify the space $$\mathcal{M }_1$$ of invariant Riemannian metrics of volume 1 on $$G/H$$ with the interior of this polytope $$N$$ . We associate with a point $${x \in \partial N}$$ of the boundary a homogeneous Riemannian space (in general, only local) and we extend the Einstein equation to $$\partial N$$ . As an application of the Alekseevsksky–Kimel’fel’d theorem, we prove that all solutions of the Einstein equation associated with points of the boundary are locally Euclidean. We describe explicitly the set $$T\subset \partial N$$ of solutions at the boundary together with its natural triangulation. Investigating the compactification $${\overline{\mathcal{M }}}_{1} = N$$ of $$\mathcal{M }_1$$ , we get an algebraic proof of the deep result by Böhm, Wang and Ziller about the compactness of the set $$\mathcal{E }_1 \subset \mathcal{M }_1$$ of Einstein metrics. The original proof by Böhm, Wang and Ziller was based on a different approach and did not use the simplicity of the spectrum. In Appendix, we consider the non-symmetric flag manifolds $$G/H$$ with the second Betti number $$b_2=1$$ . We calculate the normalized volumes $$2,6,20,82,344$$ of the corresponding Newton polytopes and discuss the number of complex solutions of the algebraic Einstein equation and the finiteness problem.
Ending Page	500
Page Count	30
Starting Page	471
File Format	PDF
ISSN	0232704X
e-ISSN	15729060
Journal	Annals of Global Analysis and Geometry
Issue Number	4
Volume Number	44
Language	English
Publisher	Springer Netherlands
Publisher Date	2013-06-23
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Geometry Einstein metric Delannoy numbers Newton polytope Homogeneous manifolds Statistics for Business/Economics/Mathematical Finance/Insurance Analysis Theoretical, Mathematical and Computational Physics Group Theory and Generalizations Homogeneous space Special Riemannian manifolds (Einstein, Sasakian, etc.)
Content Type	Text
Resource Type	Article
Subject	Analysis Geometry and Topology

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