Existence and non-existence of Einstein metrics on compact homogeneous manifolds

Abstract: We discuss existence and non-existence of Einstein metrics on certain compact homogeneous manifolds $G/H$. Study of existence and non-existence of Einstein metrics on $G/H$ is started by Wang and Ziller in 1986. Park and speaker studied some spaces $G/H$ with three irreducible summands in 1997, and more general results for non-existence are obtained by Böhm in 2005. In 2008, Dickinson and Kerr studied existence and non-existence of Einstein metrics on $G/H$ with two irreducible summands.

One of the results of Böhm is that compact homogeneous manifolds $G/H = SU(n+k_1+ \cdots +k_p)/S(SO(n)U(1)\times U(k_1) \times\cdots\times U(k_p))$ (where $SO(n)U(1) \subset U(n) $) does not admit $G$-invariant Einstein metrics, if $n > (k_1+\cdots+k_p)^2+2$. He has obtained the results by considering ``traceless" part of Ricci tensor.

We can apply his method for other compact homogeneous manifolds $G/H$.

1) For $G/H = Sp(n+k_1+ \cdots +k_p)/SO(n)Sp(1)\times Sp(k_1) \times\cdots\times Sp(k_p)$ (where $SO(n)Sp(1) \subset Sp(n)$), if $n \geq 6 (k_1+\cdots+k_p)$ ($k_i \geq 1$), then $G/H$ does not admit $G$-invariant Einstein metrics

2) For $G/H = SO(4 n+k_1+ \cdots +k_p)/Sp(n)Sp(1)\times SO(k_1) \times\cdots\times SO(k_p)$ (where $Sp(n)Sp(1) \subset SO(4n)$), if $n \geq (3/2)( k_1+ \cdots +k_p)$ ($k_i \geq 3$), then $G/H$ does not admit $G$-invariant Einstein metrics.

For $G/H = SO(2 n+k_1+ \cdots +k_p)/ SO(n)U(1)\times SO(k_1) \times\cdots\times SO(k_p)$ (where $SO(n)U(1) \subset U(n) \subset SO(2 n)$), we can not apply the method directly. In this talk, we will show that, if $n \geq 5 (k_1+ \cdots +k_p)$ ($k_i \geq 3$), $G/H$ does not admit $G$-invariant Einstein metrics by modifying argument of ``traceless" part of Ricci tensor.

For existence of Einstein metrics, in general, it is difficult for compact homogeneous manifolds $G/H$ above. But, in case of $ k_1 = \cdots = k_p=k $, we see that, for a given $p \geq 1$ there exists a pair $(n, k)$ such that $G/H$ admits at least two $G$-invariant Einstein metrics.

differential geometry

Audience: researchers in the topic

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Workshop on compact homogeneous Einstein manifolds

Series comments: The workshop aims at bringing together mathematicians that have contributed and are contributing to the study of Einstein metrics on the challenging class of all compact homogeneous spaces.

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