Rigidity of $SU_n$-type symmetric spaces

Abstract

We prove that the bi-invariant Einstein metric on SU2n+1 is isolated in the moduli space of Einstein metrics, even though it admits infinitesimal deformations. This gives a non-Kähler, non-product example of this phenomenon adding to the famous example of CP 2n × CP 1 found by Koiso. We apply our methods to derive similar solitonic rigidity results for the Kähler-Einstein metrics on 'odd' Grassmannians. We also make explicit a connection between non-integrable deformations and the dynamical instability of metrics under Ricci flow.

Coindex and rigidity of Einstein metrics on homogeneous Gray manifolds

“…Another recent example due to W. Batat, S. J. Hall, T. Murphy and J. Waldron is the bi-invariant metric on SU(2n + 1) [2]. We add one more example to this list by proving the following result.…”

“…We denote by E = C 2 the standard representation of K = SU (2). Furthermore, we label the irreducible complex representations of K by k ∈ N 0 , where V k = Sym k E is the unique (k + 1)-dimensional irreducible complex representation of K.…”

“…Buitruille [3]. There are only four cases: S 6 = G 2 SU(3) , S 3 × S 3 = SU(2)×SU( 2)×SU (2) ∆ SU (2) , CP 3 = Sp (2) Sp( 1) U(1) = SO(5) U( 2) and the flag manifold F 1,2 = SU (3) T 2 , all of them equipped with the Killing form metric (up to scaling). In [22], C. Wang and M. Y.-K. Wang show instability of the latter three spaces.…”

“…1.1 Theorem. Let (M, g) be a homogeneous Gray manifold with standard metric g. The coindex of the Einstein metric g is 2)×SU (2) ∆ SU (2) ,…”

Rigidity of complex projective spaces in Ricci shrinkers

“…Rigid examples include spherical space-forms S n /Γ [13], CP 2m by Kröncke [17], and S 2 × S 2 proved by Sun-Zhu [29] recently. For other rigid compact symmetric spaces, see [1] and those with λ −1 µ fns > 2 and H. stable in [5, Table 1, Table 2]. For noncompact Ricci shrinkers, the rigidity problem is much more involved.…”