Stability of Compact Symmetric Spaces

Abstract

In this article, we study the stability problem for the Einstein–Hilbert functional on compact symmetric spaces following and completing the seminal work of Koiso on the subject. We classify in detail the irreducible representations of simple Lie algebras with Casimir eigenvalue less than the Casimir eigenvalue of the adjoint representation and use this information to prove the stability of the Einstein metrics on both the quaternionic and Cayley projective plane. Moreover, we prove that the Einstein metrics on quaternionic Grassmannians different from projective spaces are unstable.

Coindex and rigidity of Einstein metrics on homogeneous Gray manifolds

“…for each fixed γ ∈ Ĝ. We call this mapping the prototypical differential operator associated to D and γ (as introduced by U. Semmelmann and G. Weingart in [19]). On a reductive homogeneous space, a choice of reductive complement m determines a G-invariant connection ∇ red on V M, called the canonical reductive (or Ambrose-Singer ) connection, by stipulating that…”

Stability of the Non-Symmetric Space $E_7/\mathrm{PSO}(8)$

“…which are in turn eigenspaces for (A * A) 0 for the eigenvalues 0 and 20 respectively. In order to apply (20) to the third summand, we use the plethysm Sym 2 Λ 4 = Λ 4,4 ⊕ Λ 6,2 ⊕ Λ 8,0 to decompose Sym 2 (m 1 ⊕ m 2 ) * into three irreducible subrepresentations under su( 8 ) 0…”

“…• R 8 of Weyl tensors on R 8 all remaining irreducible representations are pairwise non-isomorphic so that there is no ambiguity as to which of the two Casimir eigenvalues 80 or 56 for Cas g su (8) 0 applies in equation (20). Leaving out the calculation of the Casimir eigenvalues for Cas g so (8) 0 via Freudenthal's formula (11), we tabulate the eigenspaces and eigenvalues of the auxiliary curvature term A * A found so far:…”

“…Koiso studied in [8] the stability question for symmetric spaces. It turned out that most of the irreducible symmetric spaces of compact type are linearly stable and only very few are unstable (see also [20] and [16] for the proof in the cases not covered by Koiso). Further examples of stable Einstein metrics are provided by Einstein metrics of negative sectional curvature (see [1], Cor.…”

Coindex and Rigidity of Einstein Metrics on Homogeneous Gray Manifolds

“…for each fixed γ ∈ Ĝ. We call this mapping the prototypical differential operator associated with D and γ (as introduced by Semmelmann and Weingart [20]). On a reductive homogeneous space, a choice of reductive complement m determines a G-invariant connection ∇ red on V M, called the canonical reductive (or Ambrose-Singer) connection, by stipulating that…”