A variational approach for compact homogeneous Einstein manifolds
Let \(M = G/H\) be a homogeneous manifold of a compact Lie group \(G\) and \({\mathcal M}_1^G\) the finite dimensional non compact manifold of invariant Riemannian metrics of volume one on \(M\). It is well known that invariant Einstein metrics on \(M\) are exactly the critical points of the function \(g \mapsto S(g)\) which associates to a metric \(g \in {\mathcal M}_1^G\) its scalar curvature. The authors study conditions, under which the function \(S(g)\) satisfies the Palais-Smale condition, that is if \(g_i \in {\mathcal M}_1^G\) is a sequence such that \(S(g_i)\) is bounded and \(\| (\text{grad}S)_{g_i} \|\) converges to 0 (in \(L^2\) norm) then some subsequence of \(g_i\) converges to a critical point of \(S\). They prove theorem A: Let \(M=G/H\) be a homogeneous manifold of a compact Lie group \(G\). For any \(\varepsilon >0\) the function \(S(g)\) satisfies the Palais-Smale condition on the subset \(({\mathcal M}_1^G)_{\epsilon}= \{ g \in {\mathcal M}_1^G, \, S(g) >\varepsilon \} \). As a corollary: The moduli space of \(G\)-invariant Einstein metrics on a compact homogeneous manifold \(M = G/H\) with finite fundamental group has only finitely many components, each one of which must be compact. Theorem A is used to prove the following existence theorem. Theorem B: Let \(M = G/H\) be a homogeneous manifold of a compact Lie group \(G\). Assume that there are two connected proper closed subgroups \(K, K'\) of \(G\) which contain \(H\) and such that the homogeneous spaces \(K/H\), \(K'/H\) are not flat , i.e. admit a non flat invariant metric. Then \(M\) admits an invariant Einstein metric. In the last part, the authors study conditions, under which an invariant Einstein metric \(g\) on \(M = G/H\) is supreme that is \( S(g) = Y(M)\), where \(Y(M)\) is the Yamabe invariant, i.e. the supremum of the scalar curvatures of all Yamabe metrics on \(M\), that is metrics with smallest scalar curvature between all unit volume metrics in a conformal class. In particular, they give many examples of non supreme invariant Einstein metrics.
scientific article; zbMATH DE number 2133447