Compact Hermitian Symmetric Spaces, Coadjoint Orbits, and the Dynamical Stability of the Ricci Flow

The $G_2$ geometry of $3$-Sasaki structures

“…An elementary computation using only the sup2q-bracket relations in (1) proves ( 16) as well as pC ´2q ˝Lξ " 0. As C is self-adjoint the claim in (17) follows by duality.…”

“…Here we must have tpσq " 0 which makes that pC ´2qσ " sλ ‹ H σ after updating the last equation above. This forces d H ‹ H σ " 0 since rd H , Cs " 0 as well as L ‹ ξ σ " 0 after taking into the identity (17). Furthermore, projection onto Λ 2 H " Λ ´H ' Λ `H leads to Cσ ˘" p2 ˘sλqσ ˘which shows that σ ´P H 2´sλ .…”

“…The criterium in [27, thm.1.7], see also [25], ensures that pN 4 , g N q is dynamically unstable provided there exists f P kerp∆ g N ´24q satisfying ş N f 3 vol N ‰ 0, in which case the infinitesimal G 2 deformation εpf q is obstructed to second order. Note that on Hermitian symmetric spaces of arbitrary dimension cubic integrals for eigenfunctions of the scalar Laplacian with eigenvalue twice the Einstein constant, or equivalently Killing potentials, have been explicitly computed in [17] by the Duistermaat-Heckmann localisation formula. Based on this we obtain a new geometric proof for the G 2 rigidity of the Aloff-Wallach space, previously considered in [30,10].…”

Rigidity of $SU_n$-type symmetric spaces

“…Kröncke proved the following analogy to Lemma 2.5. eigenspace V −2λ is SU 2n+1 -equivariantly isomorphic to su 2n+1 . In [11], the authors show that there exists an eigenfunction f ∈ V −2λ with…”

“…(see [11], [13], or [20]). Hence the obstruction is a non-zero multiple of Ψ and so does not vanish for any perturbation induced by v ∈ V −2λ .…”