Annals of Global Analysis and Geometry (2022) 61:333–357
https://doi.org/10.1007/s10455-021-09810-4

Stability of Einstein metrics on symmetric spaces of compact
type

Paul Schwahn1

Received: 11 May 2021 / Accepted: 11 October 2021 / Published online: 26 November 2021
© The Author(s) 2021

Abstract
We prove the linear stability with respect to the Einstein-Hilbert action of the symmetric
spaces SU(n), n ≥ 3, and E6/F4. Combined with earlier results, this resolves the stability
problem for irreducible symmetric spaces of compact type.

Keywords Symmetric spaces · Einstein metrics · Stability · Lichnerowicz Laplacian

Mathematics Subject Classification 53C24 · 53C25 · 53C30 · 53C35

1 Introduction

Let M be a closed manifold of dimension n > 2. It is a well-known fact (see [2]) that Einstein
metrics are critical points of the total scalar curvature functional

g �→ S(g) =
∫
M
scalg volg,

also called the Einstein-Hilbert action, restricted to the space of Riemannianmetrics of a fixed
volume. In general, these critical points are neither maximal nor minimal. If we, however,
restrict S to the set S of all Riemannian metrics on M of the same fixed volume that have
constant scalar curvature, then some Einstein metrics are maximal, while others form saddle
points. To examine this, one considers the second variation S′′

g of S at a fixed Einstein metric
g on M . If we exclude the case where (M, g) is a standard sphere, the tangent space ofS at g
consists precisely of tt-tensors, i.e. symmetric 2-tensors that are transverse (divergence-free)
and traceless. In these directions, the coindex and nullity of S′′

g are always finite. The stability
problem is to decide whether they vanish for a given Einstein manifold (M, g).

The stability of an Einstein metric g is determined by the spectrum of a Laplace-type
operator ΔL , called the Lichnerowicz Laplacian, on tt-tensors. There is a critical eigenvalue,
corresponding to null directions for S′′

g , which is equal to 2E , where E is the Einstein
constant of g. The metric g is called linearly (strictly) stable if ΔL ≥ 2E (resp. ΔL > 2E)

B Paul Schwahn
paul.schwahn@mathematik.uni-stuttgart.de

1 Institut für Geometrie und Topologie, Fachbereich Mathematik, Universität Stuttgart, Pfaffenwaldring
57, 70569 Stuttgart, Germany

123

http://crossmark.crossref.org/dialog/?doi=10.1007/s10455-021-09810-4&domain=pdf
http://orcid.org/0000-0003-0718-3949


334 Annals of Global Analysis and Geometry (2022) 61:333–357

on tt-tensors, and infinitesimally deformable if there is a tt-eigentensor of ΔL for the critical
eigenvalue.

Suppose that (M, g) is a locally symmetric Einstein manifold of compact type. The
Cartan–Ambrose–Hicks theorem implies that its universal cover (M̃, g̃) is a simply connected
symmetric space. As such, (M̃, g̃) can be written as a Riemannian product of irreducible
symmetric spaces of compact type. For many of these spaces, the stability problem has
been decided by N. Koiso. The following theorem collects the results of Koiso in [10]
together with a result of J. Gasqui and H. Goldschmidt in [7] about the complex quadric
SO(5)/(SO(3) × SO(2)).

Theorem 1.1 1. The only irreducible symmetric spaces of compact type that are infinitesi-
mally deformable are

SU(n), SU(n)/SO(n), SU(2n)/Sp(n) (n ≥ 3),

SU(p + q)/S(U(p) × U(q)) (p ≥ q ≥ 2),

as well as E6/F4.
2. The irreducible symmetric spaces

Sp(n) (n ≥ 2), Sp(n)/U(n) (n ≥ 3),

as well as the complex quadric SO(5)/(SO(3) × SO(2)) are unstable.
3. Let (M, g) be an irreducible symmetric space of compact type. If (M, g) is none of the

spaces from 1. and 2., nor one of

Sp(p + q)/(Sp(p) × Sp(q)) (p ≥ q ≥ 2 or p = 2, q = 1)

nor F4/Spin(9), then g is strictly stable.

Moreover, the smallest eigenvalue of ΔL on trace-free symmetric 2-tensors has been
computed in each case (see [3]). Among the spaces that possess infinitesimal deformations,
we have ΔL ≥ 2E on S 2

0 (M) on the spaces

SU(n)/SO(n), SU(2n)/Sp(n) (n ≥ 3), SU(p + q)/S(U(p) × U(q)) (p ≥ q ≥ 2),

which shows that they are linearly stable.
However, this did not fully settle the stability problem on irreducible symmetric spaces

of compact type. In particular, it had not been decided whether unstable directions exist on
the spaces

SU(n) (where n ≥ 3), E6/F4, F4/Spin(9),

Sp(p + q)/(Sp(p) × Sp(q)) (where p ≥ q ≥ 2 or p = 2, q = 1).

In these cases, we know that ΔL has eigenvalues smaller than 2E on the space of trace-free
symmetric 2-tensors, but it had not been checked whether the corresponding eigentensors
are also divergence-free. In a recent paper [14], U. Semmelmann and G. Weingart show the
following results.

Theorem 1.2 1. The quaternionic Grassmannians Sp(p+ q)/(Sp(p) × Sp(q)) are linearly
stable for p = 2 and q = 1, but unstable for p ≥ q ≥ 2.

2. The Cayley plane OP2 = F4/Spin(9) is linearly stable.

The current article finally resolves the question of stability for the last remaining cases by
proving the following.

123



Annals of Global Analysis and Geometry (2022) 61:333–357 335

Theorem 1.3 The symmetric spaces SU(n), where n ≥ 3, as well as E6/F4 are linearly
stable.

Consider a manifold (M, g) that is a Riemannian product of Einstein manifolds. Then
(M, g) is Einstein if and only if the factors have the same Einstein constant E . It turns out
that if E > 0, then (M, g) is always unstable (see [12], Prop. 3.3.7). For example, if (M, g)
is the Riemannian product of two Einstein manifolds (Mni

i , gi ) (i = 1, 2) with the same
Einstein constant, then an unstable direction is given by

h := n2π
∗
1 g1 − n1π

∗
2 g2,

where πi : M → Mi are the projections onto each factor, respectively. In particular, a
product of symmetric spaces of compact type is always unstable since the factors have
positive curvature.

If we take (M, g) to be locally symmetric of compact type, we cannot in general conclude
its instability from the instability of its universal cover (M̃, g̃). The same holds for the exis-
tence of infinitesimal Einstein deformations. On the other hand, if (M̃, g̃) is infinitesimally
non-deformable (resp. stable), then the same follows for (M, g). In [11], N. Koiso has proved
the infinitesimal non-deformability of a large class of such manifolds:

Theorem 1.4 Let (M, g) be a locally symmetric Einstein manifold of compact type. Let
(M̃, g̃) be its universal cover and (M̃, g̃) = ∏N

i=1(Mi , gi ) its decomposition into irreducible
symmetric spaces.

1. For N = 1, see Theorem 1.1, 1.
2. If N = 2 and Mi are neither of the spaces listed in Theorem 1.1, 1., nor G2 or any

Hermitian space except S2, then (M, g) is infinitesimally non-deformable.
3. If N ≥ 3 and Mi are neither of the above nor S2, then (M, g) is infinitesimally non-

deformable.

A closely related notion of stability arises in the study of the Ricci flow. The fixed points
(modulo diffeomorphisms and scaling) of the Ricci flow are called Ricci solitons. The ν-
entropy defined by G. Perelman is a quantity that increases monotonically under the Ricci
flow. Its critical points are the shrinking gradient Ricci solitons, which include Einstein
manifolds. An Einstein metric is called ν-linearly stable if the second variation of the ν-
entropy is negative-semidefinite. H.-D. Cao, R. Hamilton and T. Ilmanen first studied the
ν-linear stability of Einstein metrics (see [4]). It turns out that an Einstein metric is ν-
linearly stable if and only if ΔL ≥ 2E on tt-tensors and if the first nonzero eigenvalue of
the ordinary Laplacian on functions is bounded below by 2E as well. In particular, ν-linear
stability implies linear stability with respect to the Einstein-Hilbert action. In [3], the ν-linear
stability of irreducible symmetric spaces of compact type is completely decided.

There is yet another notion of stability worth mentioning. It is motivated, for example,
by the investigation of Anti-de Sitter product spacetimes and generalized Schwarzschild-
Tangherlini spacetimes (see [5] or [8]). An Einstein manifold (Mn, g)with Einstein constant
E is called physically stable if

ΔL ≥ E

n − 1

(
4 − 1

4
(n − 5)2

)
= 9 − n

4
E

on tt-tensors. This critical eigenvalue is significantly smaller than the one from stability
with respect to the Einstein-Hilbert action, and even negative for n > 9. As it turns out, all
irreducible symmetric spaces of compact type are physically stable (see [5]). If (M, g) is a

123



336 Annals of Global Analysis and Geometry (2022) 61:333–357

product of at least two symmetric spaces of compact type, then the smallest eigenvalue of
ΔL on tt-tensors is actually equal to 0; hence (M, g) is physically stable if and only if n ≥ 9.

In Sect. 2, we fix the notation and definitions used throughout this work. In particular, we
elaborate on the notion of stability of an Einstein metric. In Sect. 3, we recall some tools from
the harmonic analysis of homogeneous spaces that are routinely employed. Furthermore, we
prove a technical lemma that allowsus tomake explicit computations involving the divergence
operator.Ahelpful formula for the dimension of tt-eigenspaces of theLichnerowiczLaplacian
is worked out in Sect. 4, generalizing a proposition of Koiso and utilizing properties of
Killing vector fields on Einstein manifolds. Sect. 5 uses representation theory to determine
the stability of SU(n), making use of the formula from Sect. 4; in Sect. 6, the same is done
for E6/F4. A different approach for proving the stability of both spaces that involves explicit
computations of the divergence operator can be found in the Appendix.

2 Preliminaries

Throughout what follows, let (M, g) be a compact, orientable Riemannian manifold. Let ∇
denote the Levi-Civita connection of g. The Riemannian curvature tensor, Ricci tensor and
scalar curvature are in our convention given as

R(X , Y )Z := ∇X∇Y Z − ∇Y∇X Z − ∇[X ,Y ]Z ,

Ric(X , Y ) := tr(Z �→ R(Z , X)Y ),

scal := trg Ric,

respectively.1 The action of the Riemannian curvature extends to an endomorphism on tensor
bundles as

R(X , Y ) = ∇X∇Y − ∇Y∇X − ∇[X ,Y ],

where ∇ also denotes the induced connection on the respective tensor bundle. Furthermore,
let S p(M) = Γ (Symp T ∗M) for p ≥ 0. We denote by

δ : S p+1(M) → S p(M)

the divergence operator on symmetric tensors, given by

δ = −
∑
i

ei�∇ei .

The space of tt-tensors, i.e. trace- and divergence-free symmetric 2-tensors on M , is denoted
by S 2

tt (M).
Let δ∗ : S p(M) → S p+1(M) be the formal adjoint2 of the divergence operator. It can

be written as

δ∗ =
∑
i

e�
i � ∇ei ,

where (ei ) is a local orthonormal basis of T M . Here, � denotes the (associative) symmetric
product, defined by

α � β := (k + l)!
k!l! sym(α ⊗ β)

1 We use the index g only when the metric-dependence of an object is to be emphasized.
2 That is, with respect to the inner product 〈·, ·〉g on Symp T ∗M with orthonormal basis (e�i1

� . . . � e�i p ).

123



Annals of Global Analysis and Geometry (2022) 61:333–357 337

for α ∈ Symk T , β ∈ Syml T , where T is any vector space and the symmetrization map
sym : T⊗k → Symk T is given by

sym(X1 ⊗ . . . ⊗ Xk) := 1

k!
∑
σ∈Sk

Xσ(1) ⊗ . . . ⊗ Xσ(k)

for X1, . . . , Xk ∈ T . This is analogous to the definition of the wedge product via the alter-
nation map. For tensors α, β of rank 1, we have

α � β = α ⊗ β + β ⊗ α.

It should be noted that δ∗X � = LXg for any vector field X ∈ X(M). Consequently, the
kernel of δ∗ on �1(M) is (via the metric) isomorphic to the space of Killing vector fields
on (M, g). More generally, symmetric tensors α ∈ S k(M) with δ∗α = 0 are called Killing
tensors of rank k, and δ∗ is sometimes called the Killing operator.

Definition 2.1 On tensors of any rank, the following operators are defined:

1. The curvature endomorphism q(R) is defined by

q(R) :=
∑
i< j

(ei ∧ e j )∗R(ei , e j ),

where (ei ) is a local orthonormal basis of T M and the asterisk indicates the natural action
of Λ2T ∼= so(T ).

2. The Lichnerowicz Laplacian ΔL is defined by

ΔL := ∇∗∇ + q(R).

Recall that on �p(M), p ≥ 0, this coincides with the Hodge Laplacian Δ.

On the space of Riemannian metrics on M , which is an open cone in S 2(M), the total
scalar curvature functional or Einstein-Hilbert action is given by

S(g) =
∫
M
scalg volg

for any Riemannian metric g on M . As mentioned earlier, if we restrict this functional to the
space of metrics of a fixed total volume, then Einstein metrics are precisely the critical points
of the restriction of S.

Let (M, g) be an Einstein manifold with Einstein constant E ∈ R, that is

Ric = Eg,

and suppose that (M, g) is not isometric to a standard round sphere. Denote

C∞
g (M) =

{
f ∈ C∞(M)

∣∣∣∣
∫
M

f volg = 0

}
.

It is well known (see [2]) that there is a decomposition of S 2(M), which is orthogonal
with respect to the second variation S′′

g of the total scalar curvature functional, into the four
summands

S 2(M) = Rg ⊕ C∞
g (M)g ⊕ im δ∗ ⊕ S 2

tt (M).

123



338 Annals of Global Analysis and Geometry (2022) 61:333–357

These correspond to infinitesimal changes in the metric by homothety, volume-preserving
conformal scaling, the action of diffeomorphisms, and moving within S, respectively. The
second variation S′′

g is positive on C∞
g (M)g, zero on im δ∗ and is given by

S′′
g (h, h) = −1

2
(ΔLh − 2Eh, h)g

onS 2
tt (M), where it has finite coindex and nullity; that is, the maximal subspace ofS 2

tt (M)

where S′′
g is nonnegative is finite-dimensional. In fact, the null directions in S 2

tt (M) are
precisely the infinitesimal Einstein deformations of g, i.e. infinitesimal deformations of g
that preserve the Einstein property, the total volume and are orthogonal to the orbit of g
under diffeomorphisms.

Definition 2.2 An Einstein metric g on M is called

1. (linearly) stable (with respect to the Einstein-Hilbert action) if S′′
g ≤ 0 on S 2

tt (M) or,

equivalently, if ΔL ≥ 2E on S 2
tt (M). Otherwise it is called (linearly) unstable.

2. strictly (linearly) stable (with respect to the Einstein-Hilbert action) if S′′
g < 0 onS 2

tt (M)

or, equivalently, if ΔL > 2E on S 2
tt (M).

3. infinitesimally deformable if ΔLh = 2Eh for some nonzero h ∈ S 2
tt (M).

3 Invariant differential operators

Let G be a compact Lie group with Lie algebra g and K a closed subgroup such that (M =
G/K , g) is a reductive Riemannian homogeneous space with K -invariant decomposition
g = k ⊕ m, where k is the Lie algebra of K and m is the reductive complement which is
canonically identified with the tangent space ToM at the base point o := eK ∈ M . Recall
that for some representation ρ : K → Aut V , the left-regular representation on the space of
K -equivariant smooth functions C∞(G, V )K is defined as

� : G → AutC∞(G, V )K : (�(x) f )(y) := f (x−1y)

for x, y ∈ G. Furthermore, the space C∞(G, V )K is identified with the space of sections of
the associated bundle G ×ρ V over M . The identification is given by

Γ (G ×ρ V ) → C∞(G, V )K : s �→ ŝ,

where ŝ is defined by s([x]) = [x, ŝ(x)] for any x ∈ G. If V can be expressed in terms of
the isotropy representation m, then G ×ρ V is a tensor bundle; for example, we have

X(M) = Γ (T M) ∼= Γ (G ×ρ m) ∼= C∞(G,m)K ,

�1(M) = Γ (T ∗M) ∼= Γ (G ×ρ m∗) ∼= C∞(G,m)K ,

S 2(M) = Γ (Sym2 T ∗M) ∼= Γ (G ×ρ Sym2 m∗) ∼= C∞(G,Sym2 m)K ,

S 2
0 (M) = Γ (Sym2

0 T
∗M) ∼= Γ (G ×ρ Sym2

0 m
∗) ∼= C∞(G,Sym2

0 m)K ,

where Sym2
0, S

2
0 denotes the space of trace-free elements with respect to the metric. Note

that the invariant Riemannian metric yields an equivalence between m and m∗.
Suppose that V is a complex representation. Choose a maximal torus T inside G with

Lie algebra t. Recall that up to equivalence, every irreducible finite-dimensional complex

123



Annals of Global Analysis and Geometry (2022) 61:333–357 339

representation of G is characterized by its highest weight γ ∈ t∗. By the Peter-Weyl theo-
rem and Frobenius reciprocity (cf. [15]), the left-regular representation C∞(G, V )K can be
decomposed into irreducible summands as3

C∞(G, V )K ∼=
⊕
γ

Vγ ⊗ HomK (Vγ , V ), (PW)

where γ runs over all highest weights of G-representations and (Vγ , ργ ) is the (up to
equivalence) unique irreducible representation of G with highest weight γ . For any

α ⊗ A ∈ Vγ ⊗ HomK (Vγ , V ),

the corresponding element of C∞(G, V )K is defined by

f Aα : G → V : x �→ A(ργ (x−1)α).

Since the Lichnerowicz LaplacianΔL onΓ (G×ρV ) is aG-invariant differential operator,
Schur’s Lemma implies that on each of the isotypical subspaces

Vγ ⊗ HomK (Vγ , V ),

ΔL acts as an endomorphism of the finite-dimensional vector space HomK (Vγ , V ), that is,

ΔL f Aα = f
Lγ (A)
α

for some Lγ ∈ EndHomK (Vγ , V ).
In order to obtain the spectrum of ΔL , one would have to find the eigenvalues of each Lγ

– a potentially very cumbersome task. We will shortly see that this matter is considerably
simpler in the symmetric case.

Fix an Ad-invariant inner product 〈·, ·〉g on the Lie algebra g. If we assume that G is
semisimple, one such inner product is given by −B, where B is the Killing form on g,
defined by

B(X , Y ) := tr(ad(X) ◦ ad(Y ))

for X , Y ∈ g. Recall that for any representation π : G → AutW , the Casimir operator CasGπ
with respect to the chosen inner product is an equivariant endomorphism of W , defined as

CasGπ := −
∑
i

dπ(ei ) ◦ dπ(ei )

for any orthonormal basis (ei ) of g.
The following proposition combines two well-known results that allow us to compute

the eigenvalues of ΔL on compact symmetric spaces, the latter being a formula due to H.
Freudenthal (cf. [6]).

Proposition 3.1 Let (M = G/K , g) be a compact Riemannian symmetric space where the
Riemannian metric is induced by anAd-invariant inner product 〈·, ·〉g on g, and let ρ : K →
Aut V be a representation.

1. On the left-regular representation Γ (G×ρ V ), the Lichnerowicz LaplacianΔL coincides
with the Casimir operator CasG� of the representation � : G → Aut Γ (G ×ρ V ).

3 Here, the bar over the direct sum denotes the closure in C∞(G, V )K (with the L2 inner product). In other
words,

⊕
γ Vγ ⊗ HomK (Vγ , V ) is dense in C∞(G, V )K . In fact, it is dense in L2(G, V )K , but for our

purposes, it suffices to consider smooth sections.

123



340 Annals of Global Analysis and Geometry (2022) 61:333–357

2. On each irreducible representation Vγ , the Casimir eigenvalue is given by

CasGγ = 〈γ, γ + 2δg〉t∗ ,
where δg is the half-sum of positive roots and 〈·, ·〉t∗ is the inner product on t∗ induced
by the inner product on t ⊂ g.

Remark 3.2 The first statement is a consequence of a more general result. LetG be a compact
Lie group and (M = G/K , g) be a reductive Riemannian homogeneous space. To the
reductive decomposition corresponds a canonical G-invariant connection on M (also called
the Ambrose-Singer connection), which we denote by ∇̄. This connection in turn defines a
curvature tensor R̄ and an analogue to the Lichnerowicz Laplacian via

Δ̄ := ∇̄∗∇̄ + q(R̄),

called the standardLaplacian of this connection (introduced in [13]). Then, in fact, Δ̄ = CasG�
on Γ (G ×ρ V ). The above statement follows when we note that on Riemannian symmetric
spaces, the Ambrose-Singer connection coincides with the Levi-Civita connection.

According to (PW), we canwrite the complexified left-regular representation on trace-free
symmetric 2-tensors as

S 2
0 (M)C ∼=

⊕
γ

Vγ ⊗ HomK (Vγ ,Sym2
0 m

C).

Recall that irreducible symmetric spaces of compact type can be endowedwith a Riemannian
metric induced by the Killing form (the so-called standard metric). In this case, the critical
eigenvalue ofΔL is 2E = 1. Supposing we have a representation Vγ with subcritical Casimir
eigenvalue CasGγ < 1 occurring in this decomposition, it remains to check whether the ten-
sors in the corresponding subspace are divergence-free. By Schur’s Lemma, the G-invariant
operator

δ : S 2
0 (M)C → �1(M)C

is constant on each irreducible subspace. This means that we can regard δ as a linear mapping

δ : HomK (Vγ ,Sym2
0 m

C) → HomK (Vγ ,mC),

the so-called prototypical differential operator associated to δ andVγ . For a further discussion
of invariant differential operators on homogeneous spaces, we refer the reader to Section 2
of [14].

The following lemma is of use when we need to calculate δ explicitly. A derivation of
essentially the same formula can also be found in [14], Section 2.

Lemma 3.3 Suppose (M, g) is a Riemannian symmetric space. Let h ∈ S 2(M)C correspond
to an element

α ⊗ A ∈ Vγ ⊗ HomK (Vγ ,Sym2 mC)

in the decomposition (PW) ofS 2(M)C. Let further (ei ) be an orthonormal basis ofm. Then
we have

(δh)o(X) =
∑
i

〈A(dργ (ei )α), ei � X〉

for any X ∈ m ∼= ToM.

123



Annals of Global Analysis and Geometry (2022) 61:333–357 341

Proof The element of C∞(G,Sym2 mC)K corresponding to h ∈ S 2(M) is given by

ĥ = f Aα : G → Sym2 mC : x �→ A(ργ (x−1)α),

where ργ is the representation of G on Vγ . The covariant derivative of h at the base point
may be expressed by

(∇h)o(X , Y ) = 〈dĥe, X � Y 〉
for X , Y ∈ m ∼= ToM , since ∇ coincides with the Ambrose-Singer connection on M as a
reductive homogeneous space. This implies that

(δh)o(X) = −
∑
i

ei�∇ei h(X) = −
∑
i

∇ei h(ei , X) = −
∑
i

〈dĥ(ei ), ei � X〉

= −
∑
i

〈d f Aα (ei ), ei � X〉 =
∑
i

〈A(dργ (ei )α), ei � X〉.

��

4 tt-Eigenspaces of the Lichnerowicz Laplacian

We return to the general setting of a compact Einstein manifold (M, g). Define

θ : �1(M) → S 2
0 (M) : α �→ δ∗α + 2

n
δα · g,

so that θα is precisely the trace-free part of δ∗α ∈ S 2(M). The kernel of this operator is (via
the metric) isomorphic to the space of conformal Killing fields on (M, g), that is, the space
of vector fields X ∈ X(M) such that LX g = f g for some f ∈ C∞(M). We thus call θ the
conformal Killing operator.

The following lemma is a generalization of a proposition by Koiso [11, Prop. 3.3]. For
the proof, we refer the reader to the Appendix.

Lemma 4.1 Let (M, g) be a compact Einstein manifold of dimension n ≥ 3. For any λ ∈ R,
the dimension of the eigenspace of ΔL to the eigenvalue λ on tt-tensors is given by

dim ker(ΔL − λ)
∣∣
S 2

tt (M)
= dim ker(ΔL − λ)

∣∣
S 2

0 (M)
− dim ker(Δ − λ)

∣∣
�1(M)

+ dim
(
ker(Δ − λ)

∣∣
�1(M)

∩ ker θ
)

.

At first glance, the third term on the right hand side of the above formula does not look
very amenable to computation. However, matters are made easier if we observe the following
properties of (conformal)Killing vector fields onEinsteinmanifolds, both ofwhich are proven
in the Appendix.

Lemma 4.2 On any compact Einstein manifold (M, g) not isometric to a standard round
sphere, conformalKilling fields are actuallyKilling, that is, L X g = f g for some f ∈ C∞(M)

implies f = 0. Equivalently, ker θ = ker δ∗ on �1(M).

Lemma 4.3 Any Killing field X ∈ X(M) on an Einstein manifold with Einstein constant E
satisfies

ΔX � = 2EX �.

Equivalently, ker δ∗ ⊂ ker(Δ − 2E) on �1(M).

123



342 Annals of Global Analysis and Geometry (2022) 61:333–357

If we assume that (M, g) is not isometric to a standard sphere, we can immediately
conclude that the intersection ker(Δ − λ)

∣∣
�1(M)

∩ ker θ is trivial if λ �= 2E . By virtue of
Lemma 4.1, we obtain the following.

Corollary 4.4 Let (M, g) be a compact Einstein manifold that is not isometric to a standard
round sphere, and let E be its Einstein constant. For any λ �= 2E, the dimension of the
eigenspace of ΔL to the eigenvalue λ on tt-tensors is given by

dim ker(ΔL − λ)
∣∣
S 2

tt (M)
= dim ker(ΔL − λ)

∣∣
S 2

0 (M)
− dim ker(Δ − λ)

∣∣
�1(M)

.

Remark 4.5 If we set λ = 2E in Lemma 4.1 and note that

ker(Δ − 2E)
∣∣
�1(M)

∩ ker θ = ker δ∗∣∣
�1(M)

(as Koiso did in his proof of [11, Prop. 3.3]), we recover the original formula for the critical
eigenvalue

dim ker(ΔL − 2E)
∣∣
S 2

tt (M)
= dim ker(ΔL − 2E)

∣∣
S 2

0 (M)
− dim ker(Δ − 2E)

∣∣
�1(M)

+ dim ker δ∗∣∣
�1(M)

.

Remark 4.6 Although the dimension formula of Lemma 4.1 works on any compact Einstein
manifold (M, g), it is worth mentioning that if additionally, (M, g) carries the structure of a
Riemannian homogeneous space M = G/K , the result can be refined in terms of irreducible
representations ofG. Namely, if Vγ is an irreducible representation ofG, then themultiplicity
of Vγ in the (complexified) left-regular representation on tt-tensors is given by

dimHomG(Vγ ,S 2
tt (M)C) = dimHomK (Vγ ,Sym2

0 m
C) − dimHomK (Vγ ,mC)

+ dimHomG(Vγ , (ker θ)C).

As in the proof of Lemma 4.1, the dimension formula essentially arises from the short exact
sequence

0 −→ ker θ
⊂−→ �1(M)

θ−→ S 2
0 (M)

P−→ S 2
tt (M) −→ 0

and the fact that the Laplacian commutes with every arrow. In the homogeneous case, we
note that we have a short exact sequence of G-representations and use Frobenius reciprocity
to arrive at the statement.

5 The symmetric space SU(n)

Throughout what follows, let n ≥ 3. As a symmetric space, SU(n) = G/K where G =
SU(n) × SU(n) and K = SU(n) is diagonally embedded, i.e. via

SU(n) ↪→ SU(n) × SU(n) : k �→ (k, k).

Let g and k denote the corresponding Lie algebras of G and K , respectively. We endow M
with the standard metric g induced by the Killing form on g. Hence, M is Einstein with
critical eigenvalue 2E = 1. The reductive decomposition of g with respect to g is given by

g = k̃ ⊕ m,

123



Annals of Global Analysis and Geometry (2022) 61:333–357 343

where

k̃ = {(X , X) | X ∈ k},
m = {(X ,−X) | X ∈ k}.

The K -representations k, k̃ and m are all equivalent. We denote by E = C
n the standard

representation of K .

Lemma 5.1 Let Vγ be an irreducible complex representation of G with CasGγ < 1 and

HomK (Vγ ,Sym2
0 k

C) �= 0.

Then Vγ is equivalent to one of the G-representations E ⊗ E∗ and E∗ ⊗ E. In fact,

dimHomK (Vγ ,Sym2
0 k

C) = 1

and the Casimir eigenvalue is CasGγ = (n−1)(n+1)
n2

.

Proof Let t be the torus of diagonal matrices in k. The dual t∗ is generated by the weights
ε1, . . . , εn of the defining representation E . Explicitly,

ε j (X) = X j , 1 ≤ j ≤ n

for X = diag(iX1, . . . , iXn) ∈ t. Note that ε1 + . . . + εn = 0.
Fix the ordering on roots and weights such that the simple roots of k are given by

ε j − ε j+1, 1 ≤ j ≤ n − 1.

The semigroup of dominant integral weights is then generated by the fundamental weights

ω j =
j∑

k=1

ε j , 1 ≤ j ≤ n − 1,

cf. [6, §15.1]. The highest weights of representations of K , i.e. all the dominant integral
weights, are precisely the linear combinations

γ =
n−1∑
r=1

arωr

with coefficients ar ∈ N0. The fundamental weights themselves correspond to the represen-
tations

Vωr = Λr E ∼= Λn−r E∗.

Let γ, γ ′ ∈ t∗ be two dominant integral weights. In particular, they satisfy

〈γ, γ ′〉t∗ ≥ 0.

Using Freudenthal’s formula for the Casimir operator CasKγ of a K -representation Vγ , this
implies the estimate

CasKγ+γ ′ = 〈γ + γ ′ + 2δk, γ + γ ′〉t∗ = 〈γ + 2δk, γ 〉t∗ + 2〈γ, γ ′〉t∗ + 〈γ ′ + 2δk, γ
′〉t∗

≥ 〈γ + 2δk, γ 〉t∗ + 〈γ ′ + 2δk, γ
′〉t∗ = CasKγ +CasKγ ′ .

123



344 Annals of Global Analysis and Geometry (2022) 61:333–357

In particular, we obtain

CasKγ ≥
∑
r

ar Cas
K
ωr

(∗)

for γ = ∑n−1
r=1 arωr .

The Casimir eigenvalues of the fundamental representations are given as

CasKωr
= (n + 1)r(n − r)

2n2

for r = 1, . . . , n − 1. Note that this expression is symmetric around r = n
2 and strictly

increasing for r ≤ n
2 . Furthermore, we can compute that

CasKω1
= (n + 1)(n − 1)

2n2
< 1,

CasKω2
= (n + 1)(n − 2)

n2
< 1,

CasKω3
=

⎧⎪⎨
⎪⎩

7
8 < 1, n = 6,
48
49 < 1, n = 7,
3(n+1)(n−3)

2n2
> 1, n ≥ 8,

CasKω1
+CasKω2

> 1, n ≥ 4,

CasK2ω1
> 1,

CasKω1+ωn−1
= 1,

cf. table on p. 15 of [14]. Combining the above with inequality (∗), we can deduce that if γ

is a highest weight with CasKγ < 1, then necessarily

γ ∈ {0, ω1, ωn−1, ω2, ωn−2, ω3, ωn−3︸ ︷︷ ︸
if n=6,7

}.

These dominant integral weights are, respectively, highest weights of the representations C,
E , E∗, Λ2E , Λ2E∗, Λ3E , Λ3E∗ of K .

The irreducible representations of G = K × K are precisely the tensor products of
irreducible representations of K . Let γ, γ ′ be highest weights of K -representations such
that

CasG(γ,γ ′) = CasKγ +CasKγ ′ < 1

holds. Assuming that γ, γ ′ �= 0, we conclude that γ, γ ′ ∈ {ω1, ωn−1}. This yields the four
pairwise inequivalentG-representations E⊗E , E⊗E∗, E∗ ⊗E and E∗ ⊗E∗. Furthermore,
in the case of γ = 0 or γ ′ = 0 we obtain the representations of K that were listed above,
composed with the projection onto one factor,

G → K : (k1, k2) �→ k1 or (k1, k2) �→ k2,

respectively. By restricting the mentioned G-representations to K via the embedding

K → G : k �→ (k, k),

we again obtain the irreducible K -representations C, E , E∗, Λ2E , Λ2E∗, Λ3E , Λ3E∗ as
well as the tensor product representations E ⊗ E , E ⊗ E∗ and E∗ ⊗ E∗. The latter are not

123



Annals of Global Analysis and Geometry (2022) 61:333–357 345

irreducible, but decompose into irreducible summands as follows:

E ⊗ E = Sym2 E ⊕ Λ2E,

E ⊗ E∗ = E ⊗0 E∗ ⊕ C,

E∗ ⊗ E∗ = Sym2 E∗ ⊕ Λ2E∗.

Here E ⊗0 E∗ is the set of trace-free elements of E ⊗ E∗ when regarded as n × n-matrices
over C. As a representation of K , we have

E ⊗0 E∗ ∼= Vω1+ωn−1
∼= kC.

The K -representation Sym2 kC ∼= Sym2(E ⊗0 E∗) appears on one hand as a summand
of

Sym2(E ⊗ E∗) ∼= Sym2(E ⊗0 E∗ ⊕ C) ∼= Sym2(E ⊗0 E∗) ⊕ E ⊗0 E∗ ⊕ C.

On the other hand, the symmetric power of the tensor product is given by4

Sym2(E ⊗ E∗) ∼= Sym2 E ⊗ Sym2 E∗ ⊕ Λ2E ⊗ Λ2E∗.

The tensor products Sym2 E ⊗ Sym2 E∗ and Λ2E ⊗ Λ2E∗ can in turn be decomposed into

Sym2 E ⊗ Sym2 E∗ ∼= V2ω1+2ωn−1 ⊕ Vω1+ωn−1 ⊕ C,

Λ2E ⊗ Λ2E∗ ∼=
{
E∗ ⊗ E ∼= Vω1+ωn−1 ⊕ C, n = 3,

Vω2+ωn−2 ⊕ Vω1+ωn−1 ⊕ C, n ≥ 4.

By comparing summands, we see that

Sym2(E ⊗0 E∗) ∼= V2ω1+2ωn−1 ⊕ Vω2+ωn−2︸ ︷︷ ︸
if n≥4

⊕ E ⊗0 E∗ ⊕ C.

Hence, the trace-free part is given by

Sym2
0(E ⊗0 E∗) ∼= V2ω1+2ωn−1 ⊕ Vω2+ωn−2︸ ︷︷ ︸

if n≥4

⊕ E ⊗0 E∗.

Now that we have decomposed the relevant representations into irreducible summands, we
recognize that E⊗E∗ and E∗⊗E are the only two of the specified subcritical representations
of G that, after restriction to K , have a common summand with Sym2

0 k
C. In each case, the

summand in question E ⊗0 E∗ ∼= kC appears with multiplicity 1; hence we have

dimHomK (E ⊗ E∗,Sym2
0 k

C) = dimHomK (E∗ ⊗ E,Sym2
0 k

C) = 1.

Moreover, both G-representations exhibit the same Casimir eigenvalue

CasG(ω1,ωn−1)
= CasG(ωn−1,ω1)

= CasKω1
+CasKωn−1

= (n − 1)(n + 1)

n2
.

��
According toLemma5.1, the only representations ofG (up to equivalence)with subcritical

Casimir eigenvalue that occur in decomposition (PW) ofS 2
0 (M)C are E ⊗ E∗ and E∗ ⊗ E ,

and we have

dimHomK (E ⊗ E∗,Sym2
0 m

C) = dimHomK (E∗ ⊗ E,Sym2
0 m

C) = 1

4 This is a consequence of, for example, the formula Symd (V ⊗ W ) = ⊕
Sλ(V ) ⊗ Sλ(W ) in [6, Ex. 6.11].

123



346 Annals of Global Analysis and Geometry (2022) 61:333–357

(recall thatm ∼= k), i.e. the summand occurs with multiplicity 1. It remains to check whether
the tensors in the corresponding subspaces are divergence-free. Since

E ⊗ E∗ ∼= kC ⊕ C

as a representation of K , we have

dimHomK (E ⊗ E∗,mC) = dimHomK (E∗ ⊗ E,mC) = 1,

meaning that both summands also occur in the left-regular representation �1(M) with the
same multiplicity. It now follows from Corollary 4.4 that

dim ker(ΔL − λ)
∣∣
S 2

tt (M)
= 0

for λ = (n−1)(n+1)
n2

. Since this is the only subcritical eigenvalue on S 2
0 (M), we have shown

the following.

Proposition 5.2 The symmetric space SU(n) is linearly stable.

6 The symmetric space E6/F4

Let (H, ◦) be the Albert algebra, where H is the set of Hermitian 3 × 3-matrices over the
octonions, i.e.

H :=
⎧⎨
⎩
⎛
⎝a x ȳ
x̄ b z
y z̄ c

⎞
⎠

∣∣∣∣a, b, c ∈ R, x, y, z ∈ O

⎫⎬
⎭ ,

and with Jordan multiplication defined by

X ◦ Y := 1

2
(XY + Y X).

The exceptional Lie group E6 can be realized as

E6 :=
{
α ∈ AutC HC

∣∣∣∣α preserves determinant and inner product

}
,

while F4 is defined as the set of algebra automorphisms

F4 := Aut(H, ◦).

By complex-linearly extending linear automorphisms of H, one obtains the inclusion
AutR H ⊂ AutC HC. In this sense, we have F4 ⊂ E6. In fact,

F4 = E6 ∩ AutR H.

As a representation of E6, HC is irreducible. As an F4-representation, H decomposes into
the irreducible summands

H ∼= H0 ⊕ R,

where H0 is the set of trace-free elements of H. An invariant inner product on H is defined
by

〈A, B〉 := tr(A ◦ B)

123



Annals of Global Analysis and Geometry (2022) 61:333–357 347

for A, B ∈ H. An orthogonal basis of H (cf. Section 2.1 of [16]) is given by the matrices

E1 :=
⎛
⎝1 0 0
0 0 0
0 0 0

⎞
⎠ , E2 :=

⎛
⎝0 0 0
0 1 0
0 0 0

⎞
⎠ , E3 :=

⎛
⎝0 0 0
0 0 0
0 0 1

⎞
⎠ ,

F1(x) :=
⎛
⎝0 0 0
0 0 x
0 x̄ 0

⎞
⎠ , F2(x) :=

⎛
⎝0 0 x̄
0 0 0
x 0 0

⎞
⎠ , F3(x) :=

⎛
⎝0 x 0
x̄ 0 0
0 0 0

⎞
⎠ ,

where x runs through the standard basis of O as a real vector space.
In this section, we consider the Riemannian symmetric space M = E6/F4 equipped with

the standard metric (hence with critical eigenvalue 2E = 1). The reductive decomposition
of e6 with respect to the standard metric is given by

e6 = f4 ⊕ m,

where m ∼= H0 as a representation of F4.

Lemma 6.1 Let Vγ be an irreducible complex representation of E6 with CasE6
γ < 1 and

HomF4(Vγ ,Sym2
0 H

C
0 ) �= 0.

Then Vγ is equivalent to one of the E6-representations HC and HC. In fact,

dimHomF4(H
C,Sym2

0 H
C
0 ) = dimHomF4(H

C,Sym2
0 H

C
0 ) = 1,

and the Casimir eigenvalue is CasGγ = 13
18 .

Proof We abstain from specifying a particular choice of simple root system and fundamental
weights for E6 and F4, since we are merely interested in the corresponding fundamental
representations of the respective Lie group. Following the enumerative convention of Bour-
baki (as used by the software package LiE), if we denote the fundamental weights of E6 by
ω1, . . . , ω6 and of F4 by η1, . . . , η4, then the associated representations are identified as

Vω1 = 27 ∼= HC, Vω2 = 78 ∼= eC6 , Vω3 = 351 ∼= Λ2HC,

Vω4 = 2925 ∼= Λ3HC, Vω5 = 351 ∼= Λ2HC, Vω6 = 27 ∼= HC,

Vη1 = 52 ∼= fC4 , Vη2 = 1274, Vη3 = 273, Vη4 = 26 ∼= HC

0 ,

where the number indicates the dimension.
As in the proof of Lemma 5.1, we have the estimate

CasE6
γ ≥

6∑
r=1

ar Cas
E6
ωr

for any representation Vγ of E6 with highest weight

γ =
6∑

r=1

arωr .

Among the fundamental representations, only the Casimir eigenvalues

CasE6
ω1

= CasE6
ω6

= 13

18

123



348 Annals of Global Analysis and Geometry (2022) 61:333–357

are smaller than 1 (see table on p. 16 of [14]). Since 13
18 + 13

18 > 1, it follows that only the

representations to the highest weights C,HC,HC come into question.
Consider now the F4-representation HC

0
∼= Vη4 . We obtain5 the decomposition

Sym2 Vη4
∼= V2η4 ⊕ Vη4 ⊕ C

into irreducible summands, hence

Sym2
0 H

C
0

∼= V2η4 ⊕ HC
0 .

Furthermore, we have

HC ∼= HC ∼= HC
0 ⊕ C

as a representation of F4. The assertion follows by comparison of summands. ��
Lemma 6.1 now tells us that the representations of E6 with subcritical Casimir eigenvalue

that occur in decomposition (PW)ofS 2
0 (M)C are preciselyHC andHC, bothwithmultiplicity

1, i.e.

dim HomF4(H
C,Sym2

0 m
C) = dimHomF4(H

C,Sym2
0 m

C) = 1,

since m ∼= H0. Again, we have to check whether the tensors in the corresponding subspace
are divergence-free. It follows from the decomposition H = H0 ⊕ R as a representation of
F4 that

dim HomF4(H
C,mC) = dimHomF4(H

C,mC) = 1,

so as in the previous section, the summand has the same multiplicity in the left-regular
representation �1(M). Again, it follows from Corollary 4.4 that

dim ker(ΔL − λ)
∣∣
S 2

tt (M)
= 0

for λ = 13
18 , and since this is the only subcritical eigenvalue on S 2

0 (M), we have shown the
following, which, together with Proposition 5.2, finishes the proof of the main theorem.

Proposition 6.2 The symmetric space E6/F4 is linearly stable.

Funding Open Access funding enabled and organized by Projekt DEAL. No funds, grants, or other support
was received.

Data availability statement Data sharing not applicable to this article as no datasetswere generated or analysed
during the current study.

Declarations

Employment During the time of research as well as currently, the author is employed as a research assistant
at the University of Stuttgart.

Competing interests The author has no relevant financial or non-financial interests to disclose.

Code availability Not applicable.

5 This has been verified through use of the software package LiE v2.1. See, for example, http://young.sp2mi.
univ-poitiers.fr/cgi-bin/form-prep/marc/sym-alt.act?x1=0&x2=0&x3=0&x4=1&power=2&kind=sym&
rank=4&group=F4 or enter the command sym_tensor(2,[0,0,0,1],F4) into the LiE shell.

123

http://young.sp2mi.univ-poitiers.fr/cgi-bin/form-prep/marc/sym-alt.act?x1=0&x2=0&x3=0&x4=1&power=2&kind=sym&rank=4&group=F4
http://young.sp2mi.univ-poitiers.fr/cgi-bin/form-prep/marc/sym-alt.act?x1=0&x2=0&x3=0&x4=1&power=2&kind=sym&rank=4&group=F4
http://young.sp2mi.univ-poitiers.fr/cgi-bin/form-prep/marc/sym-alt.act?x1=0&x2=0&x3=0&x4=1&power=2&kind=sym&rank=4&group=F4


Annals of Global Analysis and Geometry (2022) 61:333–357 349

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give
appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence,
and indicate if changes were made. The images or other third party material in this article are included in the
article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is
not included in the article’s Creative Commons licence and your intended use is not permitted by statutory
regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

A Appendix

A.1 Proofs of general statements

Proof of Lemma 4.1 The following is a slightly generalized version of the proof of a result by
N. Koiso [11, Prop. 3.3]. We first note that

(δα · g, h)g =
∫
M

δα〈g, h〉g volg = 1

2
(δα, trg h)g = 1

2
(α, d trg h)g

for α ∈ �1(M), h ∈ S 2(M), so the formal adjoint of θ is given by

θ∗ : S 2
0 (M) → �1(M) : h �→ δh + 1

n
d trg h.

We show that θ is overdetermined elliptic. The principal symbol of θ is

σξ (θ)α = σξ (δ
∗)α + 2

n
σξ (δ)α · g = ξ � α − 2

n
〈ξ, α〉gg

for ξ, α ∈ T ∗
p M . If ξ �= 0, then σξ (θ) is injective: Suppose σξ (θ)α = 0. Then

ξ � α = 2

n
〈ξ, α〉gg.

Take an orthonormal basis (ei ) with respect to g of TpM and write

ξ =
∑
i

ξi e
�
i , α =

∑
i

αi e
�
i .

For i, j = 1, . . . , n, it follows that

ξiα j + ξ jαi = 2

n
〈ξ, α〉gδi j

and so ξiα j = −ξ jαi if i �= j , as well as ξiαi = ξ jα j for any i, j . Then

ξ2i α j = −ξiαiξ j = −ξ2j α j .

If α j �= 0, this would imply that ξ2i + ξ2j = 0 and so ξi = ξ j = 0, which contradicts the
assumption that ξ �= 0. Overall, we conclude that α = 0 and thus the injectivity is proven.

From ellipticity, we obtain the orthogonal decomposition

S 2
0 (M) = im θ ⊕ ker θ∗.

Let h ∈ ker(ΔL − λ)
∣∣
S 2

0 (M)
. According to the above decomposition, we can write h as

h = θα + ψ

123

http://creativecommons.org/licenses/by/4.0/


350 Annals of Global Analysis and Geometry (2022) 61:333–357

where θ∗ψ = 0. Then also,

δψ = θ∗ψ − 1

n
d trg ψ = 0.

Since (M, g) is Einstein, ΔL commutes with δ on S 2(M) and with δ∗ on �1(M) [9,
10.7/10.8]. Furthermore ΔL( f g) = (Δ f )g for any f ∈ C∞(M). We conclude that ΔL

commutes with θ and θ∗ as well. This implies that

θ(Δ − λ)α = (ΔL − λ)θα = (ΔL − λ)(h − ψ) = −(ΔL − λ)ψ,

θ∗(ΔL − λ)ψ = (Δ − λ)θ∗ψ = 0,

and so

θ∗θ(Δ − λ)α = −θ∗(ΔL − λ)ψ = 0.

It follows that

‖θ(Δ − λ)α‖2g = (
θ∗θ(Δ − λ)α, (Δ − λ)α

)
g = 0

and so θ(Δ − λ)α = 0 = (ΔL − λ)ψ . In total, ψ ∈ ker(ΔL − λ)
∣∣
S 2

tt (M)
.

Also, if h is an element of ker(ΔL − λ)
∣∣
S 2

tt (M)
, then

θ∗h = δh + 1

n
d trg h = 0

and so ψ = h. This means that the mapping

P : ker(ΔL − λ)
∣∣
S 2

0 (M)
→ ker(ΔL − λ)

∣∣
S 2

tt (M)
: h �→ ψ

defines a projection, and the dimension formula

dim ker(ΔL − λ)
∣∣
S 2

tt (M)
= dim

(
ker(ΔL − λ)

∣∣
S 2

0 (M)

)
− dim ker P

holds.
By definition, the kernel of P consists of those h ∈ ker(ΔL − λ)

∣∣
S 2

0 (M)
with h = θα for

some α ∈ �1(M), i.e. h ∈ im θ . Hence we know that

ker P = ker(ΔL − λ)
∣∣
S 2

0 (M)
∩ im θ.

Let α ∈ ker(Δ − λ)
∣∣
�1(M)

. We have seen that ΔL commutes with θ , so it follows that

θα ∈ ker(ΔL − λ)
∣∣
S 2

0 (M)
and therefore

θ
(
ker(Δ − λ)

∣∣
�1(M)

)
⊂ ker P.

Conversely, let h ∈ ker P . Then there exists some α ∈ �1(M) such that h = θα, and also
h ∈ ker(Δg

L − λ)
∣∣
S 2

0 (M)
. By the ellipticity of the operator Δ − λ, we can decompose α into

α = β + (Δ − λ)γ

123



Annals of Global Analysis and Geometry (2022) 61:333–357 351

with β ∈ ker(Δ − λ)
∣∣
�1(M)

, γ ∈ �1(M). Then

0 = (ΔL − λ)θα

= (ΔL − λ)θβ + (ΔL − λ)θ(Δ − λ)γ

= θ(Δ − λ)β + (ΔL − λ)2θγ

= (ΔL − λ)2θγ.

Since ΔL is self-adjoint, we have

‖(ΔL − λ)θγ ‖2g = (
(ΔL − λ)2θγ, θγ

)
g = 0

and thus

θ(Δ − λ)γ = (ΔL − λ)θγ = 0,

i.e. (Δ − λ)γ ∈ ker θ . This implies that h = θα = θβ, so

θ : ker(Δ − λ)
∣∣
�1(M)

→ ker P

is surjective and we obtain the dimension formula

dim ker P = dim ker(Δ − λ)
∣∣
�1(M)

− dim
(
ker(Δ − λ)

∣∣
�1(M)

∩ ker θ
)

.

��

Proof of Lemma 4.2 Let E be the Einstein constant of (M, g). Let α ∈ �1(M) such that

θα = δ∗α + 2

n
δα · g = 0.

Taking the divergence yields

δθα = δδ∗α − 2

n
dδα = 0,

since δ( f g) = −d f for f ∈ C∞(M).Wemake use of thewell-knownWeitzenböck identities

δδ∗ − δ∗δ = ∇∗∇ − q(R) on S k(M),

Δ = d∗d + dd∗ = ∇∗∇ + q(R) on �k(M).

For k = 1 and since δ∗ = d = ∇ on functions and (M, g) is Einstein, these amount to

δδ∗α − dδα =∇∗∇α − Eα,

d∗dα + dδα =∇∗∇α + Eα.

Putting these together, we obtain

δθα =
(
1 − 2

n

)
dδα + ∇∗∇α − Eα =

(
2 − 2

n

)
dδα + d∗dα − 2Eα = 0.

Taking the L2 inner product with α then yields
(
2 − 2

n

)
‖δα‖2g + ‖dα‖2g − 2E‖α‖g = 0.

123



352 Annals of Global Analysis and Geometry (2022) 61:333–357

If E < 0, this directly implies that α = 0. If E = 0, it implies δα = 0 and dα = 0, and since
θα = 0, it follows that δ∗α = 0. If E > 0, then applying the codifferential to δθα yields(

2 − 2

n

)
d∗dδα + (d∗)2dα − 2Ed∗α =

(
2 − 2

n

)
Δδα − 2Eδα = 0,

so δα would be an eigenfunction of the Laplacian to the eigenvalue En
n−1 = scalg

n−1 . By a
theorem of Obata [1, Thm. D.I.6], this eigenvalue can only be attained on the standard
sphere, so necessarily δα = 0. It follows again from θα = 0 that δ∗α = 0. ��
Proof of Lemma 4.3 Let α ∈ �1(M) such that δ∗α = 0. Then also δα = 0, since δα =
− trg δ∗α = 0. By virtue of the Weitzenböck formulae that were already employed in the
proof of Lemma 4.2, we conclude that

Δα = ∇∗∇α + Eα = δδ∗α − dδα + 2Eα = 2Eα.

��

A.2 Alternative proof of the stability of SU(n)

An alternative method of checking that the prototypical differential operators

δ : HomK (E ⊗ E∗,Sym2
0 m

C) → HomK (E ⊗ E∗,mC),

δ : HomK (E∗ ⊗ E,Sym2
0 m

C) → HomK (E∗ ⊗ E,mC)

are injective is an explicit computation by means of Lemma 3.3. To do so, we first pick out
an explicit element

A ∈ HomK (E ⊗ E∗,Sym2
0 m

C)

and then proceed to compute the divergence on the corresponding subspace of S 2
0 (M).

Lemma A.1 Let π : Sym2(E ⊗ E∗) → E ⊗ E∗ denote the mapping defined by

π(A � B) := AB∗ + BA∗,

where A, B ∈ E ⊗ E∗ are regarded as complex n × n-matrices. Then

π ∈ HomK (Sym2(E ⊗ E∗), E ⊗ E∗).

Moreover, the restriction

π : Sym2
0(E ⊗0 E∗) → E ⊗0 E∗

is surjective, and W :=
(
ker π

∣∣
Sym2

0(E⊗0E∗)

)⊥ ∼= E ⊗0 E∗.

Proof The equivariance of π under the action of K follows from

π(k Ak−1 � kBk−1) = k Ak−1(k−1)∗B∗k∗ + k−1Bk(k−1)∗A∗k∗ = k(AB∗ + BA∗)k−1

for any k ∈ K = SU(n) and A, B ∈ E ⊗ E∗. Furthermore, we have

tr(π(A � B)) = tr(AB∗ + BA∗) = 〈A, B〉 + 〈B, A〉 = tr(A � B),

where the last trace is taken with respect to the inner product on E ⊗ E∗. This means that

π(Sym2
0(E ⊗ E∗)) ⊂ E ⊗0 E∗.

123



Annals of Global Analysis and Geometry (2022) 61:333–357 353

Next wewant to show that π does not vanish when restricted to Sym2
0(E⊗0 E∗). If we denote

by Ei j the n × n-matrix that has entry 1 at position (i, j) and 0 elsewhere, then we have for
example E21, E31 ∈ E ⊗0 E∗ and 〈E21, E31〉 = 0, so E21 � E31 ∈ Sym2

0(E ⊗0 E∗) and

π(E21 � E31) = E21E13 + E31E12 = E23 + E32 �= 0.

Now, since E ⊗0 E∗ is irreducible, the mapping

π : Sym2
0(E ⊗0 E∗) → E ⊗0 E∗

must be surjective. We have seen in the proof of Lemma 5.1 that E ⊗0 E∗ appears in the

decomposition of Sym2
0(E ⊗0 E∗) with multiplicity 1; hence W :=

(
ker π

∣∣
Sym2

0(E⊗0E∗)

)⊥

must be the irreducible summand of Sym2
0(E ⊗0 E∗) that is equivalent to E ⊗0 E∗. ��

Alternative proof of Proposition 5.2 The properties of π from Lemma A.1 allow us to define

à := π
∣∣−1
W ∈ HomK (E ⊗0 E∗,Sym2

0(E ⊗0 E∗))

and extend it with zero to a mapping à ∈ HomK (E ⊗ E∗,Sym2
0(E ⊗0 E∗)). Via the

identification mC ∼= E ⊗0 E∗, this gives rise to a mapping

A ∈ HomK (E ⊗ E∗,Sym2
0 m

C).

From the equivariance of π
∣∣
W , the irreducibility of W ∼= E ⊗0 E∗ and Schur’s Lemma it

follows that π
∣∣
W is unitary up to a positive constant, that is

〈π(v), π(w)〉E⊗0E∗ = c · 〈v,w〉Sym2
0(E⊗0E∗)

for all v,w ∈ W and some c > 0. Denote the tensor product representation of G on E ⊗ E∗
by

ρ : G → Aut(E ⊗ E∗) : ρ(k1, k2)F = k1Fk
−1
2

for F ∈ E ⊗ E∗. Its differential is given by

dρ : g → End(E ⊗ E∗) : dρ(X1, X2)F = X1F − FX2

for X1, X2 ∈ k. In particular,

dρ(X ,−X)F = XF + FX .

Let (ei ) be an orthonormal basis of m, ei = ( fi ,− fi ) with fi ∈ k. Under the identification
mC ∼= E ⊗0 E∗, the invariant inner product changes by some positive constant factor, and ei
is mapped to fi . Hence, ( fi ) is an orthonormal basis of k ⊂ E ⊗0 E∗ up to a positive factor.

Now, let X ∈ k and F ∈ E ⊗ E∗. Using the formula from Lemma 3.3, it follows that

(δh)o(X ,−X) =
∑
i

〈A(dρ(ei )F), ei � (X ,−X)〉Sym2
0 m

C

= c ·
∑
i

〈 Ã( fi F + F fi ), fi � X〉Sym2
0(E⊗0E∗)

= c ·
∑
i

〈 Ã( fi F + F fi ), prW ( fi � X)〉Sym2
0(E⊗0E∗)

= c′ ·
∑
i

〈 fi F + F fi , π(prSym2
0(E⊗0E∗)( fi � X))〉E⊗0E∗

123



354 Annals of Global Analysis and Geometry (2022) 61:333–357

for some c, c′ > 0. Since the trivial summand of Sym2(E ⊗0 E∗) can only be mapped to the
trivial summand of E ⊗ E∗ under the equivariant map π , we have

π ◦ prSym2
0(E⊗0E∗) = prE⊗0E∗ ◦ π

on Sym2(E ⊗0 E∗), implying that

(δh)o(X ,−X) = c′ ·
∑
i

〈 fi F + F fi , prE⊗0E∗( fi X
∗ + X f ∗

i )〉

= −c′ ·
∑
i

〈 fi F + F fi , prE⊗0E∗( fi X + X fi )〉.

Choose the (up to a positive factor) orthonormal basis ( fi ) of k in such a way that f1 =
E12 − E21. Furthermore, let X = F = E13 − E31. Then,

f1F + F f1 = (E12 − E21)(E13 − E31) + (E13 − E31)(E12 − E21) = −E23 − E32 ∈ E ⊗0 E∗

and we obtain∑
i

〈 fi F + F fi , prE⊗0E∗( fi X + X fi )〉 =
∑
i

〈 fi F + F fi , prE⊗0E∗( fi F + F fi )〉

≥ 〈 f1F + F f1, prE⊗0E∗( f1F + F f1)〉 = 〈E23 + E32, E23 + E32〉 = 2 > 0.

In particular, we have found Y ∈ m such that (δh)o(Y ) �= 0, where h ∈ S 2
0 (M) is associated

to

F ⊗ A ∈ (E ⊗ E∗) ⊗ HomK (E ⊗ E∗,Sym2
0 m

C).

This means that the linear mapping

δ : HomK (E ⊗ E∗,Sym2
0 m

C) → HomK (E ⊗ E∗,mC)

is nonzero. Hence, there are no tt-eigentensors for the subcritical Casimir eigenvalue. This
proves the assertion. ��

A.3 Alternative proof of the stability of E6/F4

As we did before in the situation of SU(n), we want to apply Lemma 3.3 to verify that the
mappings

δ : HomF4(H
C,Sym2

0 m
C) → HomF4(H

C,mC),

δ : HomF4(H
C,Sym2

0 m
C) → HomF4(H

C,mC)

are injective. Surprisingly, the computation works very similar to the SU(n) case.

Lemma A.2 Let π : Sym2 H → H denote the mapping defined by

π(A � B) := AB + BA = 2A ◦ B.

Then we have

π ∈ HomF4(Sym
2 H0,H).

The restriction

π : Sym2
0 H0 → H0

123



Annals of Global Analysis and Geometry (2022) 61:333–357 355

is surjective, and W :=
(
ker π

∣∣
Sym2

0 H0

)⊥ ∼= H0.

Proof The proof is completely analogous to the proof of Lemma A.1. First, we note that π is
well-defined since (H, ◦) is a commutative algebra. The equivariance of π under the action
of F4 follows from

π( f (A) � f (B)) = 2 f (A) ◦ f (B) = f (2A ◦ B) = f (π(A � B))

for any f ∈ F4 = Aut(H, ◦) and A, B ∈ H. Furthermore, we have

tr(π(A � B)) = 2 tr(A ◦ B) = 2〈A, B〉 = tr(A � B),

where the last trace is taken with respect to the inner product on H. This means that

π(Sym2
0 H) ⊂ H0.

Now we want to show that π does not vanish when restricted to Sym2
0 H0. For example, take

F1(1), F2(1) ∈ H0. We have 〈F1(1), F2(1)〉 = 0 and thus F1(1) � F2(1) ∈ Sym2
0 H0. Also,

π(F1(1) � F2(1)) = 2F1(1) ◦ F2(1) = F3(1) �= 0.

Since H0 is irreducible over F4, the mapping

π : Sym2
0 H0 → H0

must be surjective. From the proof of Lemma 6.1, we know that H0 appears in the decompo-

sition of Sym2
0 H0 with multiplicity 1; henceW :=

(
ker π

∣∣
Sym2

0 H0

)⊥
must be the irreducible

summand of Sym2
0 H0 that is equivalent to H0. ��

Alternative proof of Proposition 6.2 By Lemma A.2, we can define

A := π
∣∣−1
W ∈ HomF4(H0,Sym

2
0 H0),

extend it with zero toH and then complex-linearly to a mapping A ∈ HomF4(H
C,Sym2

0 H
C

0 ).
Again, we need thatπ

∣∣
W is unitary up to a positive constant, which follows by Schur’s Lemma

from the equivariance of π
∣∣
W and the irreducibility of W ∼= H0. By Theorem 3.2.4 in [16],

every element α ∈ e6 ⊂ EndC(HC) can be written as

α = β + iT ◦
with unique elements β ∈ f4 ⊂ e6 and T ∈ H0. This corresponds to the F4-invariant
decomposition

e6 ∼= f4 ⊕ H0.

Throughout what follows, we identify m ∼= H0. If we denote the defining representation by

ρ : E6 → AutHC,

then in particular,

dρ(X) = iX◦

123



356 Annals of Global Analysis and Geometry (2022) 61:333–357

for X ∈ m. Let (ei ) be an orthonormal basis of H0 (again, under the identification m ∼= H0,
the invariant inner product changes at most by some positive constant factor), X ∈ m and
F ∈ HC. Using Lemma 3.3, we thus obtain

(δh)o(X) =c ·
∑
i

〈A(dρ(ei )F), ei � X〉Sym2
0 H

C
0

= c ·
∑
i

〈A(iei ◦ F), ei � X〉Sym2
0 H

C
0

= c ·
∑
i

〈A(iei ◦ F), prW (ei � X)〉Sym2
0 H

C
0

=c′ ·
∑
i

〈iei ◦ F, π(prSym2
0 H0

(ei � X))〉HC
0

for some c, c′ > 0. The trivial summand of Sym2 H0 can only be mapped to the trivial
summand of H under the equivariant map π , implying that

π ◦ prSym2
0 H0

= prH0
◦π

on Sym2 H0. Thus, we have

(δh)o(X) = ic′ ·
∑
i

〈ei ◦ F, prH0
(π(ei � X))〉 = 2ic′ ∑

i

〈ei ◦ F, prH0
(ei ◦ X)〉.

Now let X = F = F1(1). Choose the (up to a positive factor) orthonormal basis (ei ) of
H0 in such a way that e1 = F2(1). Then we have

e1 ◦ F = F2(1) ◦ F1(1) = 1

2
F3(1) ∈ H0

and it follows that∑
i

〈ei ◦ F, prH0
(ei ◦ X)〉 =

∑
i

〈ei ◦ F, prH0
(ei ◦ F)〉 ≥ 〈e1 ◦ F, prH0

(e1 ◦ F)〉

= 1

4
〈F3(1), F3(1)〉 = 1

2
> 0.

In particular, we have found Y ∈ m such that (δh)o(Y ) �= 0, where h ∈ S 2
0 (M) is associated

to

F ⊗ A ∈ HC ⊗ HomF4(H
C,Sym2

0 m
C).

This means that the linear mapping

δ : HomF4(H
C,Sym2

0 m
C) → HomF4(H

C,mC)

is nonzero. The same argumentworks for the E6-representationHC, sincewe exclusively used
real elements and automorphisms in the computation. In total, there are no tt-eigentensors
for the subcritical Casimir eigenvalue, which proves the assertion. ��

References

1. Berger, M., Gauduchon, P., Mazet, E.: Le spectre d’une variété Riemannienne, Lecture Notes in Mathe-
matics, Vol. 194. Springer-Verlag Berlin Heidelberg (1971)

2. Besse, A.L.: Einstein Manifolds, Classics in Mathematics. Springer-Verlag Berlin Heidelberg (1987)

123



Annals of Global Analysis and Geometry (2022) 61:333–357 357

3. Cao, H.-D., He, C.: Linear stability of Perelman’s ν-entropy on symmetric spaces of compact type. J.
Reine Angew. Math. 709, 229–246 (2015)

4. Huai-Dong, C., Hamilton, R.S., Ilmanen, T.: Gaussian densities and stability for some Ricci solitons.
arXiv:math/0404165 (2004)

5. Dieterich, P.-S.: On the Lichnerowicz Laplace operator and its application to stability of spacetimes
(thesis), Universität Stuttgart (2012)

6. Fulton, W., Harris, J.: Representation Theory: A First Course, Graduate Texts in Mathematics, vol. 129.
Springer-Verlag, New York (1991)

7. Gasqui, J., Goldschmidt, H.: Radon transforms and spectral rigidity on the complex quadrics and the real
Grassmannians of rank two. J. Reine Angew. Math. 480, 1–69 (1996)

8. Gibbons, G.W., Hartnoll, S.A., Pope, C.N.: Bohm and Einstein-Sasaki metrics, black holes, and cosmo-
logical event horizons. Phys. Rev. D 67(8), 084024 (2003)

9. Lichnerowicz, A.: Propagateurs et commutateurs en relativité générale. Public. Mathématiques de
l’I.H.É.S. 10, 5–56 (1961)

10. Koiso, N.: Rigidity and stability of Einstein metrics: the case of compact symmetric spaces. Osaka Math.
J. 17, 51–73 (1980)

11. Koiso, N.: Rigidity and infinitesimal deformability of Einsteinmetrics. OsakaMath. J. 19, 643–668 (1982)
12. Kröncke, K.: Stability of Einstein Manifolds (thesis), University of Potsdam (2013)
13. Semmelmann, U., Weingart, G.: Manuscripta Mathematica. The standard Laplacian 158, 273–293 (2019)
14. Semmelmann, U., Weingart, G.: Linear stability of symmetric spaces. arXiv:2012.07328 (2020)
15. Wallach, N.R.: Harmonic Analysis on Homogeneous Spaces. Marcel Dekker, Inc, New York (1973)
16. Yokota, I: Exceptional Lie groups. arXiv:0902.0431 (2009)

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and
institutional affiliations.

123

http://arxiv.org/abs/math/0404165
http://arxiv.org/abs/2012.07328
http://arxiv.org/abs/0902.0431

	Stability of Einstein metrics on symmetric spaces of compact type
	Abstract
	1 Introduction
	2 Preliminaries
	3 Invariant differential operators
	4 tt-Eigenspaces of the Lichnerowicz Laplacian
	5 The symmetric space SU(n)
	6 The symmetric space E6/F4
	A Appendix
	A.1 Proofs of general statements
	A.2 Alternative proof of the stability of SU(n)
	A.3 Alternative proof of the stability of E6/F4

	References