Rigidity of the compact irreducible symmetric space

Question

Let $(M^n,g)$ be an irreducible symmetric space of compact type. In particular, $(M^n,g)$ is an Einstein manifold with a positive Einstein constant.

Is there any classification for $(M^n,g)$ if $(M^n,g)$ is rigid in the Einstein manifolds? Here, $(M^n,g)$ is rigid if, whenever $(M^n,g')$ is another Einstein manifold with $g'$ close to $g$ in $C^2$ sense, then $g=g'$ up to a diffeomorphism and rescaling.

I know that Koiso showed in Rigidity and stability of Einstein metrics—the case of compact symmetric spaces that all compact irreducible symmetric spaces are rigid except possibly for $$ SU(p+q)/S(U(p) \times U(q)),p\ge q \ge2; SU(m)/SO(m); SU(2m)/Sp(m);SU(m), m \ge 3;E_6/F_4. $$

Are there any results including the above cases?

Answer 1

The quite recent preprint “Rigidity of $SU(n)$-type symmetric spaces” by Batat, Hall, Murphy, and Waldron continues Koiso's study. They proved that a bi-invariant metric on $SU(n)$ is rigid when $n$ is odd. Check Subsection 1.4 for previous results; they mention that Gasqui and Goldchmidt made geometric constructions of infinitesimal deformations in some cases.

Following this link you can find a talk by Murphy about this paper.