# The Hermitian Laplace Operator on Nearly Kähler Manifolds

**ABSTRACT** The moduli space NK of infinitesimal deformations of a nearly Kähler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1,1) forms. Using the Hermitian Laplace operator and some representation theory, we compute the space NK on all 6-dimensional homogeneous nearly Kähler manifolds. It turns out that the nearly Kähler structure is rigid except for the flag manifold F(1,2)=SU_3/T^2, which carries an 8-dimensional moduli space of infinitesimal nearly Kähler deformations, modeled on the Lie algebra su_3 of the isometry group.

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**ABSTRACT:**We study generalized Killing spinors on the standard sphere $\mathbb{S}^3$, which turn out to be related to Lagrangian embeddings in the nearly K\"ahler manifold $S^3\times S^3$ and to great circle flows on $\mathbb{S}^3$. Using our methods we generalize a well known result of Gluck and Gu concerning divergence-free geodesic vector fields on the sphere and we show that the space of Lagrangian submanifolds of $S^3\times S^3$ has at least three connected components.Differential Geometry and its Applications. 05/2014; 37. - SourceAvailable from: Jason Dean Lotay
##### Article: Deformation theory of G_2 conifolds

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**ABSTRACT:**We consider the deformation theory of asymptotically conical (AC) and of conically singular (CS) G_2 manifolds. In the AC case, we show that if the rate of convergence nu to the cone at infinity is generic in a precise sense and lies in the interval (-4, -5/2), then the moduli space is smooth and we compute its dimension in terms of topological and analytic data. For generic rates nu < -4 in the AC case, and for generic positive rates of convergence to the cones at the singular points in the CS case, the deformation theory is in general obstructed. We describe the obstruction spaces explicitly in terms of the spectrum of the Laplacian on the link of the cones on the ends, and compute the virtual dimension of the moduli space. We also present several applications of these results, including: the local uniqueness of the Bryant--Salamon AC G_2 manifolds; the smoothness of the CS moduli space if the singularities are modeled on particular G_2 cones; and the proof of existence of a "good gauge" needed for desingularization of CS G_2 manifolds. Finally, we discuss some open problems for future study.12/2012; - SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**We consider some natural infinitesimal Einstein deformations on Sasakian and 3-Sasakian manifolds. Some of these are infinitesimal deformations of Killing spinors and further some integrate to actual Killing spinor deformations. In particular, on 3-Sasakian 7 manifolds these yield infinitesimal Einstein deformations preserving 2, 1, or none of the 3 independent Killing spinors. Toric 3-Sasakian manifolds provide non-trivial examples with integrable deformation preserving precisely 2 Killing spinors. Thus the dimension of Killing spinors is not preserved under Einstein deformations but is only upper semi-continuous.01/2013;

Page 1

arXiv:0810.0164v2 [math.DG] 14 Dec 2009

THE HERMITIAN LAPLACE OPERATOR ON NEARLY K¨AHLER

MANIFOLDS

ANDREI MOROIANU AND UWE SEMMELMANN

Abstract. The moduli space NK of infinitesimal deformations of a nearly K¨ ahler

structure on a compact 6-dimensional manifold is described by a certain eigenspace of

the Laplace operator acting on co-closed primitive (1,1) forms (c.f. [10]). Using the

Hermitian Laplace operator and some representation theory, we compute the space

NK on all 6-dimensional homogeneous nearly K¨ ahler manifolds. It turns out that the

nearly K¨ ahler structure is rigid except for the flag manifold F(1,2) = SU3/T2, which

carries an 8-dimensional moduli space of infinitesimal nearly K¨ ahler deformations,

modeled on the Lie algebra su3of the isometry group.

2000 Mathematics Subject Classification: Primary 58E30, 53C10, 53C15.

Keywords: Nearly K¨ ahler deformations, Hermitian Laplace operator.

1. Introduction

Nearly K¨ ahler manifolds were introduced in the 70’s by A. Gray [8] in the context of

weak holonomy. More recently, 6-dimensional nearly K¨ ahler manifolds turned out to be

related to a multitude of topics among which we mention: Spin manifolds with Killing

spinors (Grunewald), SU3-structures, geometries with torsion (Cleyton, Swann), stable

forms (Hitchin), or super-symmetric models in theoretical physics (Friedrich, Ivanov).

Up to now, the only sources of compact examples are the naturally reductive 3-

symmetric spaces, classified by Gray and Wolf [13], and the twistor spaces over positive

quaternion-K¨ ahler manifolds, equipped with the non-integrable almost complex struc-

ture. Based on previous work by R. Cleyton and A. Swann [6], P.-A. Nagy has shown

in 2002 that every simply connected nearly K¨ ahler manifold is a Riemannian product of

factors which are either of one of these two types, or 6-dimensional [12]. Moreover,

J.-B. Butruille has shown [5] that every homogeneous 6-dimensional nearly K¨ ahler

manifold is a 3-symmetric space G/K, more precisely isometric with S6= G2/SU3,

S3×S3= SU2×SU2×SU2/SU2, CP3= SO5/U2×S1or F(1,2) = SU3/T2, all endowed

with the metric defined by the Killing form of G.

A method of finding new examples is to take some homogeneous nearly K¨ ahler man-

ifold and try to deform its structure. In [10] we have studied the deformation problem

for 6-dimensional nearly K¨ ahler manifolds (M6,g) and proved that if M is compact,

Date: December 15, 2009.

This work was supported by the French-German cooperation project Procope no. 17825PG.

1

Page 2

2ANDREI MOROIANU AND UWE SEMMELMANN

and has normalized scalar curvature scalg = 30, then the space NK of infinitesimal

deformations of the nearly K¨ ahler structure is isomorphic to the eigenspace for the

eigenvalue 12 of the restriction of the Laplace operator ∆gto the space of co-closed

primitive (1,1)-forms Λ(1,1)

0

M.

It is thus natural to investigate the Laplace operator on the known 3-symmetric exam-

ples (besides the sphere S6, whose space of nearly K¨ ahler structures is well-understood,

and isomorphic to SO7/G2∼= RP7, see [7] or [5, Prop. 7.2]). Recall that the spectrum

of the Laplace operator on symmetric spaces can be computed in terms of Casimir

eigenvalues using the Peter-Weyl formalism. It turns out that a similar method can be

applied in order to compute the spectrum of a modified Laplace operator¯∆ (called the

Hermitian Laplace operator) on 3-symmetric spaces. This operator is SU3-equivariant

and coincides with the usual Laplace operator on co-closed primitive (1,1)-forms. The

space of infinitesimal nearly K¨ ahler deformations is thus identified with the space of

co-closed forms in Ω(1,1)

0

(12) := {α ∈ C∞(Λ(1,1)

that the nearly K¨ ahler structure is rigid on S3×S3and CP3, and that the space of in-

finitesimal nearly K¨ ahler deformations of the flag manifold F(1,2) is eight-dimensional.

0

M) |¯∆α = 12α}. Our main result is

The paper is organized as follows. After some preliminaries on nearly K¨ ahler mani-

folds, we give two general procedures for constructing elements in Ω(1,1)

vector fields or eigenfunctions of the Laplace operator for the eigenvalue 12 (Corollary

4.5 and Proposition 4.11). We show that these elements can not be co-closed, thus

obtaining an upper bound for the dimension of the space of infinitesimal nearly K¨ ahler

deformations (Proposition 4.12). We then compute this upper bound explicitly on the

3-symmetric examples and find that it vanishes for S3× S3and CP3, which therefore

have no infinitesimal nearly K¨ ahler deformation. This upper bound is equal to 8 on the

flag manifold F(1,2) = SU3/T2and in the last section we construct an explicit isomor-

phism between the Lie algebra of the isometry group su3and the space of infinitesimal

nearly K¨ ahler deformations on F(1,2).

0

(12) out of Killing

In addition, our explicit computations (in Section 5) of the spectrum of the Hermitian

Laplace operator on the 3-symmetric spaces, together with the results in [11] show that

every infinitesimal Einstein deformation on a 3-symmetric space is automatically an

infinitesimal nearly K¨ ahler deformation.

Acknowledgments. We are grateful to Gregor Weingart for helpful discussions and in

particular for suggesting the statement of Lemma 5.4.

2. Preliminaries on nearly K¨ ahler manifolds

An almost Hermitian manifold (M2m,g,J) is called nearly K¨ ahler if

(∇XJ)(X) = 0,

∀ X ∈ TM,(1)

Page 3

THE HERMITIAN LAPLACE OPERATOR ON NEARLY K¨AHLER MANIFOLDS3

where ∇ denotes the Levi-Civita connection of g. The canonical Hermitian connection

¯∇, defined by

¯∇XY := ∇XY −1

is a Umconnection on M (i.e.¯∇g = 0 and¯∇J = 0) with torsion¯TXY = −J(∇XJ)Y .

A fundamental observation, which goes back to Gray, is the fact that¯∇¯T = 0 on every

nearly K¨ ahler manifold (see [2]).

We denote the K¨ ahler form of M by ω := g(J.,.). The tensor Ψ+:= ∇ω is totally

skew-symmetric and of type (3,0) + (0,3) by (1). From now on we assume that the

dimension of M is 2m = 6 and that the nearly K¨ ahler structure is strict, i.e. (M,g,J)

is not K¨ ahler. It is well-known that M is Einstein in this case. We will always normalize

the scalar curvature of M to scal = 30, in which case we also have |Ψ+|2= 4 point-wise.

The form Ψ+can be seen as the real part of a¯∇-parallel complex volume form Ψ++iΨ−

on M, where Ψ−= ∗Ψ+is the Hodge dual of Ψ+. Thus M carries a SU3structure whose

minimal connection (cf. [6]) is exactly¯∇. Notice that Hitchin has shown that a SU3

structure (ω,Ψ+,Ψ−) is nearly K¨ ahler if and only if the following exterior system holds:

2J(∇XJ)Y,

∀ X ∈ TM, ∀ Y ∈ C∞(M) (2)

?

dω = 3Ψ+

dΨ−= −2ω ∧ ω.

(3)

Let A ∈ Λ1M⊗EndM denote the tensor AX:= J(∇XJ) = −Ψ+

the endomorphism associated to Y ?Ψ+via the metric. Since for every unit vector X,

AXdefines a complex structure on the 4-dimensional space X⊥∩ (JX)⊥, we easily get

in a local orthonormal basis {ei} the formulas

|AX|2= 2|X|2,

AeiAei(X) = −4X,

where here and henceforth, we use Einstein’s summation convention on repeating sub-

scripts. The following algebraic relations are satisfied for every SU3structure (ω,Ψ+)

on TM (notice that we identify vectors and 1-forms via the metric):

JX, where Ψ+

Ydenotes

∀ X ∈ TM.

∀ X ∈ TM,

(4)

(5)

AXei∧ ei?Ψ+= −2X ∧ ω,

X ?Ψ−= −JX ?Ψ+,

(X ?Ψ+) ∧ Ψ+= X ∧ ω2,

(JX ?Ψ+) ∧ ω = X ∧ Ψ+,

∀ X ∈ TM.

∀ X ∈ TM,

∀ X ∈ TM.

∀ X ∈ TM.

(6)

(7)

(8)

(9)

The Hodge operator satisfies ∗2= (−1)pon ΛpM and moreover

∗(X ∧ Ψ+) = JX ?Ψ+,

∗(φ ∧ ω) = −φ,

∗(JX ∧ ω2) = −2X,

∀ X ∈ TM.

∀ φ ∈ Λ(1,1)

∀ X ∈ TM.

(10)

0

M. (11)

(12)

Page 4

4ANDREI MOROIANU AND UWE SEMMELMANN

From now on we assume that (M,g) is compact 6-dimensional not isometric to the

round sphere (S6,can). It is well-known that every Killing vector field ξ on M is an

automorphism of the whole nearly K¨ ahler structure (see [10]). In particular,

Lξω = 0,LξΨ+= 0,LξΨ−= 0.(13)

Let now R and¯R denote the curvature tensors of ∇ and¯∇. Then the formula (c.f. [1])

RWXY Z

=

¯RWXY Z−1

+3

4g(Y, W)g(X, Z) +1

4g(X, Y )g(Z, W)

4g(Y, JW)g(JX, Z) −3

4g(Y, JX)g(JW, Z) −1

2g(X, JW)g(JY, Z)

may be rewritten as

RXY = −X ∧ Y + RCY

XY

and

¯RXY = −3

4(X ∧ Y + JX ∧ JY −2

3ω(X,Y )J) + RCY

XY

where RCY

XYis a curvature tensor of Calabi-Yau type.

We will recall the definition of the curvature endomorphism q(R) (c.f. [10]). Let EM

be the vector bundle associated to the bundle of orthonormal frames via a representation

π : SO(n) → Aut(E). The Levi-Civita connection of M induces a connection on EM,

whose curvature satisfies REM

the differential of π and identify the Lie algebra of SO(n), i.e. the skew-symmetric

endomorphisms, with Λ2. In order to keep notations as simple as possible, we introduce

the notation π∗(A) = A∗. The curvature endomorphism q(R) ∈ End(EM) is defined as

q(R) =1

XY= π∗(RXY) = π∗(R(X ∧ Y )), where we denote with π∗

2(ei∧ ej)∗R(ei∧ ej)∗

(14)

for any local orthonormal frame {ei}. In particular, q(R) = Ric on TM. By the same

formula we may define for any curvature tensor S, or more generally any endomorphism

S of Λ2TM, a bundle morphism q(S). In any point q : R ?→ q(R) defines an equivariant

map from the space of algebraic curvature tensors to the space of endomorphisms of E.

Since a Calabi-Yau algebraic curvature tensor has vanishing Ricci curvature, q(RCY) = 0

holds on TM. Let R0

direct calculation gives

XYbe defined by R0

XY= X ∧Y +JX ∧JY −2

3ω(X,Y )J. Then a

q(R0) =1

2

?

(ei∧ ej)∗(ei∧ ej)∗+1

2

?

(ei∧ ej)∗(Jei∧ Jej)∗−2

3ω∗ω∗.

We apply this formula on TM. The first summand is exactly the SO(n)-Casimir, which

acts as −5id. The third summand is easily seen to be2

acts as −id (c.f. [11]). Altogether we obtain q(R0) = −16

expression for q(¯R) acting on TM:

3id, whereas the second summand

3id, which gives the following

q(¯R)|TM= 4idTM.(15)

Page 5

THE HERMITIAN LAPLACE OPERATOR ON NEARLY K¨AHLER MANIFOLDS5

3. The Hermitian Laplace operator

In the next two sections (M6,g,J) will be a compact nearly K¨ ahler manifold with

scalar curvature normalized to scalg= 30. We denote as usual by ∆ the Laplace operator

∆ = d∗d+dd∗= ∇∗∇+q(R) on differential forms. We introduce the Hermitian Laplace

operator

¯∆ =¯∇∗¯∇ + q(¯R),

which can be defined on any associated bundle EM. In [11] we have computed the

difference of the operators ∆ and¯∆ on a primitive (1,1)-form φ:

(∆ −¯∆)φ = (Jd∗φ)?Ψ+.

In particular, ∆ and¯∆ coincide on co-closed primitive (1,1)-forms. We now compute

the difference ∆−¯∆ on 1-forms. Using the calculation in [11] (or directly from (15)) we

have q(R) − q(¯R) = id on TM. It remains to compute the operator P = ∇∗∇ −¯∇∗¯∇

on TM. A direct calculation using (5) gives for every 1-form θ

P(θ) = −1

= −θ − Aei∇eiθ.

In order to compute the last term, we introduce the metric adjoint α : Λ2M → TM

of the bundle homomorphism X ∈ TM ?→ X?Ψ+∈ Λ2M. It is easy to check that

α(X?Ψ+) = 2X (c.f. [10]). Keeping in mind that A is totally skew-symmetric, we

compute for an arbitrary vector X ∈ TM

?Aei(∇eiθ),X? = ?AXei,∇eiθ? = ?AX,ei∧ ∇eiθ? = ?AX,dθ?

= −?Ψ+

whence Aei(∇eiθ) = Jα(dθ). Summarizing our calculations we have proved the following

Proposition 3.1. Let (M6,g,J) be a nearly K¨ ahler manifold with scalar curvature

normalized to scalg= 30. Then for any 1-form θ it holds that

(∆ −¯∆)θ = −Jα(dθ).

The next result is a formula for the commutator of J and α ◦ d on 1-forms.

Lemma 3.2. For all 1-forms θ, the following formula holds:

(16)

(17)

4AeiAeiθ − Aei¯∇eiθ = θ − Aei¯∇eiθ = θ +1

2AeiAeiθ − Aei∇eiθ

JX,dθ? = −?JX,α(dθ)? = ?Jα(dθ),X?,

α(dθ) = 4Jθ + Jα(dJθ).

Proof. Differentiating the identity θ∧Ψ+= Jθ∧Ψ−gives dθ∧Ψ+= dJθ∧Ψ−+2Jθ∧ω2.

With respect to the SU3-invariant decomposition Λ2M = Λ(1,1)M ⊕ Λ(2,0)+(0,2)M, we

can write dθ = (dθ)(1,1)+1

wedge product of forms of type (1,1) and (3,0) vanishes we derive the equation

2α(dθ)?Ψ+and dJθ = (dJθ)(1,1)+1

2α(dJθ)?Ψ+. Since the

1

2(α(dθ)?Ψ+) ∧ Ψ+=1

2(α(dJθ)?Ψ+) ∧ Ψ−+ 2Jθ ∧ ω2.

Using (8) and (9) we obtain

1

2α(dθ) ∧ ω2=1

2Jα(dJθ) ∧ ω2+ 2Jθ ∧ ω2.

Page 6

6ANDREI MOROIANU AND UWE SEMMELMANN

Taking the Hodge dual of this equation and using (12) gives Jα(dθ) = −α(dJθ) − 4θ,

which proves the lemma.

?

Finally we note two interesting consequences of Proposition 3.1 and Lemma 3.2.

Corollary 3.3. For any closed 1-form θ it holds that

(∆ −¯∆)θ = 0,

Proof. For a closed 1-form θ Lemma 3.1 directly implies that ∆ and¯∆ coincide on θ.

For the second equation we use Proposition 3.1 together with Lemma 3.2 to conclude

(∆ −¯∆)Jθ = −Jα(dJθ) = 4Jθ − α(dθ) = 4Jθ

since θ is closed. This completes the proof of the corollary.

(∆ −¯∆)Jθ = 4Jθ.

?

4. Special¯∆-eigenforms on nearly K¨ ahler manifolds

In this section we assume moreover that (M,g) is not isometric to the standard sphere

(S6,can). In the first part of this section we will show how to construct¯∆-eigenforms

on M starting from Killing vector fields.

Let ξ be a non-trivial Killing vector field on (M,g), which in particular implies d∗ξ = 0

and ∆ξ = 2Ric(ξ) = 10ξ. As an immediate consequence of the Cartan formula and (13)

we obtain

dJξ = Lξω − ξ?dω = −3ξ?ψ+

so by (4), the square norm of dJξ (as a 2-form) is

(18)

|dJξ|2= 18|ξ|2.(19)

In [9] we showed already that the vector field Jξ is co-closed if ξ is a Killing vector field

and has unit length. However it turns out that this also holds more generally.

Proposition 4.1. Let ξ be a Killing vector field on M. Then d∗Jξ = 0.

Proof. Let dv denote the volume form of (M,g). We start with computing the L2-norm

of d∗Jξ.

?d∗Jξ?2

L2

=

?

?

?

M?d∗Jξ,d∗Jξ?dv =?

M[|∇Jξ|2+ 5|ξ|2− |dJξ|2]dv =?

M[?∆Jξ,Jξ? − ?d∗dJξ,Jξ?]dv

=

M[?∇∗∇Jξ,Jξ? + 5|Jξ|2− |dJξ|2]dv

=

M[|∇Jξ|2− 13|ξ|2]dv

Here we used the well-known Bochner formula for 1-forms, i.e. ∆θ = ∇∗∇θ + Ric(θ),

with Ric(θ) = 5θ in our case. Next we consider the decomposition of ∇Jξ into its

symmetric and skew-symmetric parts 2∇Jξ = dJξ + LJξg, which together with (19)

leads to

|∇Jξ|2=1

4(|dJξ|2+ |LJξg|2) = 9|ξ|2+1

4|LJξg|2.(20)

Page 7

THE HERMITIAN LAPLACE OPERATOR ON NEARLY K¨AHLER MANIFOLDS7

(Recall that the endomorphism square norm of a 2-form is twice its square norm as a

form). In order to compute the last norm, we express LJξg as follows:

LJξg(X,Y ) = g(∇XJξ,Y ) + g(X,∇YJξ)

= g(J∇Xξ,Y ) + g(X,J∇Yξ) + Ψ+(X,ξ,Y ) + Ψ+(Y,ξ,X)

= −g(∇Xξ,JY ) − g(JX,∇Yξ) = −dξ(1,1)(X,JY ),

whence

?LJξg?2

On the other hand, as an application of Lemma 3.2 together with Equation (18) we get

α(dξ) = 4Jξ + Jα(dJξ) = −2Jξ, so

dξ(2,0)= −Jξ ?Ψ+.

Moreover, ∆ξ = 10ξ since ξ is a Killing vector field, which yields

L2 = 2?dξ(1,1)?2

L2.(21)

(22)

?dξ(1,1)?2

L2 = ?dξ?2

L2 − ?dξ(2,0)?2

L2 = 10?ξ?2

L2 − 2?ξ?2

L2 = 8?ξ?2

L2= 13?ξ?2

L2.

This last equation, together with (20) and (21) gives ?∇Jξ?2

this into the first equation proves that d∗Jξ has vanishing L2-norm and thus that Jξ is

co-closed.

L2. Substituting

?

Proposition 4.2. Let ξ be a Killing vector field on M. Then

∆ξ = 10ξ, and∆Jξ = 18Jξ.

In particular, Jξ can never be a Killing vector field.

Proof. The first equation holds for every Killing vector field on an Einstein manifold

with Ric = 5id. From (18) we know dJξ = −3ξ?Ψ+. Hence the second assertion follows

from:

d∗dJξ = − ∗ d ∗ dJξ

(10)

= −3 ∗ d(Jξ ∧ Ψ+) = 9 ∗ (ξ ∧ ω2)

(12)

= 18Jξ.

?

Since the differential d commutes with the Laplace operator ∆, every Killing vector

field ξ defines two ∆-eigenforms of degree 2:

∆dJξ = 18dJξand∆dξ = 10dξ

As a direct consequence of Proposition 4.2, together with formulas (18), (22), and

Proposition 3.1 we get:

Corollary 4.3. Every Killing vector field on M satisfies

¯∆ξ = 12ξ,

¯∆Jξ = 12Jξ.

Our next goal is to show that the (1,1)-part of dξ is a¯∆ -eigenform. By (22) we have

dξ = φ − Jξ?Ψ+, (23)

Page 8

8ANDREI MOROIANU AND UWE SEMMELMANN

for some (1,1)-form φ. Using Proposition 4.1, we can write in a local orthonormal basis

{ei}:

?dξ,ω? =1

thus showing that φ is primitive. The differential of φ can be computed from the Cartan

formula:

= −d(ξ?Ψ−) = −LξΨ−+ξ?dΨ−(13)

From here we obtain

∗dφ = −4 ∗ (Jξ ∧ ω) = 4ξ ∧ ω,

whence

4dξ ∧ ω − 12ξ ∧ Ψ+(23)

(9)

=4φ ∧ ω − 16ξ ∧ Ψ+.

Using (10) and (11), we thus get

d∗dφ = − ∗ d ∗ dφ = 4φ + 16Jξ?Ψ+.

On the other hand,

2?dξ,ei∧ Jei? = ?∇eiξ,Jei? = d∗Jξ = 0,

dφ

(23)

= d(Jξ?Ψ++dξ)

(7)

= −2ξ?ω2= −4Jξ ∧ω. (24)

d ∗ dφ = = 4φ ∧ ω − 4(Jξ?Ψ+) ∧ ω − 12ξ ∧ Ψ+

d∗φ = − ∗ d ∗ φ

(11)

= ∗d(φ ∧ ω)

(24)

= X(−4Jξ ∧ ω2+ 3φ ∧ Ψ+)

(12)

= 8ξ

and finally

dd∗φ = 8dξ = 8φ − 8Jξ?Ψ+.

The calculations above thus prove the following proposition

Proposition 4.4. Let (M6,g,J) be a compact nearly K¨ ahler manifold with scalar cur-

vature scalg= 30, not isometric to the standard sphere. Let ξ be a Killing vector field on

M and let φ be the (1,1)-part of dξ. Then φ is primitive, i.e. φ = (dξ)(1,1)

d∗φ = 8ξ and ∆φ = 12φ + 8Jξ?Ψ+.

0

. Moreover

Corollary 4.5. The primitive (1,1)-form ϕ satisfies

¯∆φ = 12φ.

Proof. From (17) and the proposition above we get

¯∆φ = ∆φ − (∆ −¯∆)φ = 12φ + 8Jξ?Ψ+− (Jd∗φ)?Ψ+= 12φ.

?

In the second part of this section we will present another way of obtaining primitive

¯∆-eigenforms of type (1,1), starting from eigenfunctions of the Laplace operator. Let

f be such an eigenfunction, i.e.∆f = λf. We consider the primitive (1,1)-form

η := (dJdf)(1,1)

0

.

Lemma 4.6. The form η is explicitly given by

η = dJdf + 2df?Ψ++λ

3fω.

Page 9

THE HERMITIAN LAPLACE OPERATOR ON NEARLY K¨AHLER MANIFOLDS9

Proof. According to the decomposition of Λ2M into irreducible SU3-summands, we can

write

dJdf = η + γ?Ψ++ hω

for some vector field γ and function h. From Lemma 3.2 we get 2γ = α(dJdf) = −4df.

In order to compute h, we write

6hdv = hω ∧ ω2= dJdf ∧ ω2= d(Jdf ∧ ω2)

(12)

= 2d ∗ df = 2λf dv.

?

We will now compute the Laplacian of the three summands of η separately. First,

we have ∆df = λdf and Corollary 3.3 yields¯∆df = λdf. Since¯∆ commutes with J, we

also have¯∆Jdf = λJdf and from the second equation in Corollary 3.3 we obtain

∆Jdf =¯∆Jdf + (∆ −¯∆)Jdf = (λ + 4)Jdf.

Hence, dJdf is a ∆-eigenform for the eigenvalue λ + 4.

Lemma 4.7. The co-differential of the (1,1)-form η is given by

d∗η =?2λ

3− 4?Jdf.

Proof. Notice that d∗(fω) = −df?ω and that d∗Jdf = −∗d∗Jdf = −1

since dω2= 0. Using this we obtain

2∗d(df ∧ω2) = 0,

d∗η = ∆Jdf + 2d∗(df?Ψ+) −λ

= (λ + 4 −λ

3df?ω = (λ + 4)Jdf − 2 ∗ d(df ∧ Ψ−) −λ

3)Jdf − 4 ∗ (df ∧ ω2)

3Jdf

(12)

= (2λ

3− 4)Jdf.

?

In order to compute ∆ of the second summand of η we need three additional formulas

Lemma 4.8.

¯∆(X?Ψ+) = (¯∆X)?Ψ+.

Proof. Recall that¯∆ =¯∇∗¯∇ + q(¯R). Since Ψ+is¯∇-parallel we immediately obtain

¯∇∗¯∇(X?Ψ+) = −¯∇ei¯∇ei(X?Ψ+) = −(¯∇ei¯∇eiX)?Ψ+.

The map A ?→ A∗Ψ+is a SU3-equivariant map from Λ2to Λ3. But since Λ3does not

contain the representation Λ(1,1)

0

as an irreducible summand, it follows that A∗Ψ+= 0

for any skew-symmetric endomorphism A corresponding to some primitive (1,1)-form.

Hence we conclude

q(¯R)(X?Ψ+) = ωi∗¯R(ωi)∗(X?Ψ+) = (ωi∗¯R(ωi)∗X)?Ψ+= (q(¯R)X)?Ψ+,

where, since the holonomy of¯∇ is included in SU3, the sum goes over some orthonormal

basis {ωi} of Λ(1,1)

0

M. Combining these two formulas we obtain¯∆(X?Ψ+) = (¯∆X)?Ψ+.

?

Page 10

10ANDREI MOROIANU AND UWE SEMMELMANN

Lemma 4.9.

(∆ −¯∆)(df?Ψ+) = 6(df?Ψ+) −4λ

3fω − 2η.

Proof. From Proposition 3.4 in [11] we have

(∆ −¯∆)(df?Ψ+) = (∇∗∇ −¯∇∗¯∇)(df?Ψ+) + (q(R) − q(¯R))(df?Ψ+)

= (∇∗∇ −¯∇∗¯∇)(df?Ψ+) + 4df?Ψ+.

The first part of the right hand side reads

(∇∗∇ −¯∇∗¯∇)(df?Ψ+) = −1

From (5) we get

4Aei∗Aei∗df?Ψ+− Aei∗¯∇ei(df?Ψ+). (25)

Aei∗Aei∗df?Ψ+

= Aei∗(Aeiek∧ Ψ+(df,ek,·))

= AeiAeiek∧ Ψ+(df,ek,·) + Aeiek∧ AeiΨ+(df,ek,·)

= −4ek∧ ek?Ψ+

where we used the vanishing of the expression E = Aeiek∧ AeiejΨ+(df,ek,ej):

E = AJeiek∧ AJeiejΨ+(df,ek,ej) = AeiJek∧ AeiJejΨ+(df,ek,ej)

= Aeiek∧ AeiejΨ+(df,Jek,Jej) = −E.

It remains to compute the second term in (25). We notice that by Schur’s Lemma, every

SU3-equivariant map from the space of symmetric tensors Sym2M to TM vanishes, so

in particular (since ∇df is symmetric), one has Aei∇eidf = 0. We then compute

Aei∗¯∇eiΨ+

(6)

=(Aei∇eidf)?Ψ+−1

=2Ψ+

2Ψ+

2Ψ+

df+ Aeiek∧ AeiejΨ+(df,ek,ej) = −8Ψ+

df,

df

=Aei∗((¯∇eidf)?Ψ+) = (Aei¯∇eidf)?Ψ++ (¯∇eidf)?Aei∗Ψ+

2(AeiAeidf)?Ψ+− 2(¯∇eidf)?(ei∧ ω)

df+ 2d∗dfω + ?Aeidf,ei?ω + 2ei∧ J¯∇eidf

df+ 2λfω + 2ei∧¯∇eiJdf = 2Ψ+

df+ 2λfω + 2dJdf + 2AJdf= 4Ψ+

=

df+ 2λfω + 2dJdf − ei∧ AeiJdf

df+ 2λfω + 2dJdf.

=

Plugging back what we obtained into (25) yields

(∇∗∇ −¯∇∗¯∇)(df?Ψ+) = −(2Ψ+

df+ 2λfω + 2dJdf),

which together with Lemma 4.6 and the first equation prove the desired formula.

?

Lemma 4.10.

∆fω = (λ + 12)fω − 2(df?Ψ+).

Proof. Since d∗(fω) = −df?ω = −Jdf we have dd∗(fω) = −dJdf. For the second

summand of ∆(fω) we first compute d(fω) = df∧ω+3fΨ+. Since d∗Ψ+=1

3d∗dω = 4ω,

Page 11

THE HERMITIAN LAPLACE OPERATOR ON NEARLY K¨AHLER MANIFOLDS11

we get d∗fΨ+= −df?Ψ++ fd∗Ψ+= −df?Ψ++ 4fω. Moreover

d∗(df ∧ ω) = − ∗ d(Jdf ∧ ω) = − ∗ (dJdf ∧ ω − 3Jdf ∧ Ψ+)

= − ∗ ([η − 2df?Ψ+−λ

= η + 2 ∗ ((df?Ψ+) ∧ ω) +2λ

= η + 2df ?Ψ++2λ

3fω] ∧ ω) + 3 ∗ (Jdf ∧ Ψ+)

3fω − 3df?Ψ+

3fω − 3df?Ψ+.

3fω, we obtain

3fω = (λ + 12)fω − 2df?Ψ+.

Recalling that η = dJdf + 2df?Ψ++λ

∆fω = −dJdf − 3df?Ψ++ 12fω + η − df?Ψ++2λ

?

Applying these three lemmas we conclude

∆(df?Ψ+) =¯∆(df?Ψ+) + (∆ −¯∆)(df?Ψ+) = (λ + 6)(df?Ψ+) −4λ

and thus

∆η = (λ + 4)dJdf + (2λ + 12)(df?Ψ+) −8λ

= λη +?4 −2λ

Finally we have once again to apply the formula for the difference of ∆ and¯∆ on

primitive (1,1)-forms. We obtain

¯∆η = ∆η − Jd∗η?Ψ+= ∆η +?2λ

Summarizing our calculations we obtain the following result.

Proposition 4.11. Let f be an ∆-eigenfunction with ∆f = λf Then the primitive

(1,1)-form η := (dJdf)(1,1)

0

satisfies

¯∆η = λη and

3fω − 2η

3fω − 4η +λ

3(λ + 12)fω −2λ

3(df?Ψ+)

3

?(df?Ψ+).

3− 4?(df?Ψ+) = λη.

d∗η =?2λ

3− 4?Jdf.

Let Ω0(12) ⊂ C∞(M) be the¯∆-eigenspace for the eigenvalue 12 (notice that¯∆ = ∆

on functions) and let Ω(1,1)

0

(12) denote the space of primitive (1,1)-eigenforms of¯∆

corresponding to the eigenvalue 12. Summarizing Corollary 4.5 and Proposition 4.11,

we have constructed a linear mapping

Φ : i(M) → Ω(1,1)

from the space of Killing vector fields into Ω(1,1)

Ψ : Ω0(12) → Ω(1,1)

Let moreover NK ⊂ Ω(1,1)

by [10] is just the space of co-closed forms in Ω(1,1)

0

(12),Φ(ξ) := dξ(1,1)

0

0

(12) and a linear mapping

Ψ(f) := (dJdf)(1,1)

0

(12),

0

.

0

(12) denote the space of nearly K¨ ahler deformations, which

(12).

0

Proposition 4.12. The linear mappings Φ and Ψ defined above are injective and the

sum Im(Φ) + Im(Ψ) + NK ⊂ Ω(1,1)

dim(NK) ≤ dim(Ω(1,1)

0

(12) is a direct sum. In particular,

0

(12)) − dim(i(M)) − dim(Ω0(12)).(26)

Page 12

12ANDREI MOROIANU AND UWE SEMMELMANN

Proof. It is enough to show that if ξ ∈ i(M), f ∈ Ω0(12) and α ∈ NK satisfy

dξ(1,1)

0

+ (dJdf)(1,1)

then ξ = 0 and f = 0. We apply d∗to (27). Using Propositions 4.4 and 4.11 to express

the co-differentials of the first two terms we get

0

+ α = 0,(27)

8ξ + 8Jdf = 0. (28)

Since Jξ is co-closed (Proposition 4.1), formula (28) implies 0 = d∗Jξ = d∗df = 12f,

i.e. f = 0. Plugging back into (28) yields ξ = 0 too.

?

5. The homogeneous Laplace operator on reductive homogeneous

spaces

5.1. The Peter-Weyl formalism. Let M = G/K be a homogeneous space with com-

pact Lie groups K ⊂ G and let π : K → Aut(E) be a representation of K. We denote

by EM := G ×πE be the associated vector bundle over M. The Peter-Weyl theorem

and the Frobenius reciprocity yield the following isomorphism of G-representations:

L2(EM)∼=

γ∈ˆG

whereˆG is the set of (non-isomorphic) irreducible G-representations. If not otherwise

stated we will consider only complex representations. Recall that the space of smooth

sections C∞(EM) can be identified with the space C∞(G;E)Kof K-invariant E-valued

functions, i.e. functions f : G → E with f(gk) = π(k)−1f(g). This space is made

into a G-representation by the left-regular representation ℓ, defined by by (ℓ(g)f)(a) =

f(g−1a). Let v ∈ Vγ and A ∈ HomK(Vγ,E) then the invariant E-valued function

corresponding to v ⊗A is defined by g ?→ A(g−1v). In particular, each summand in the

Hilbert space direct sum (29) is a subset of C∞(EM) ⊂ L2(EM).

Let g be the Lie algebra of G. We denote by B the Killing form of g, B(X,Y ) :=

tr(adX◦adY). The Killing form is non-degenerated and negative definite if G is compact

and semi-simple, which will be the case in all examples below.

?

Vγ⊗ HomK(Vγ,E),(29)

If π : G → Aut(E) is a G-representation, the Casimir operator of (G,π) acts on E

by the formula

CasG

?

where {Xi} is a (−B)-orthonormal basis of g and π∗: g → End(E) denotes the differ-

ential of the representation π.

π=(π∗Xi)2, (30)

Remark 5.1. Notice that the Casimir operator is divided by k if one use the scalar

product −kB instead of −B.

If G is simple, the adjoint representation ad on the complexification gCis irreducible,

so, by Schur’s Lemma, its Casimir operator acts as a scalar. Taking the trace in (30)

for π = ad yields the useful formula CasG

ad= −1.

Page 13

THE HERMITIAN LAPLACE OPERATOR ON NEARLY K¨AHLER MANIFOLDS13

Let Vγ be an irreducible G-representation of highest weight γ. By Freudenthal’s

formula the Casimir operator acts on Vγby scalar multiplication with ?ρ?2− ?ρ+ γ?2,

where ρ denotes the half-sum of the positive roots and ?·? is the norm induced by −B

on the dual of the Lie algebra of the maximal torus of G. Notice that these scalars are

always non-positive. Indeed ?ρ?2− ?ρ + γ?2= −?γ,γ + 2ρ?Band ?γ,ρ? ≥ 0, since γ

is a dominant weight, i.e. it is in the the closure of the fixed Weyl chamber, whereas

ρ is the half-sum of positive weights and thus by definition has a non-negative scalar

product with γ.

5.2. The homogeneous Laplace operator. We denote by¯∇ the canonical homoge-

neous connection on M = G/K. It coincides with the Levi-Civita connection only in

the case that G/K is a symmetric space. A crucial observation is that the canonical

homogeneous connection coincides with the canonical Hermitian connection on nat-

urally reductive 3-symmetric spaces (see below). We define the curvature endomor-

phism q(¯R) ∈ End(EM) as in (14) and introduce as in (16) the second order operator

¯∆π=¯∇∗¯∇ + q(¯R) acting on sections of the associated bundle EM := G ×πE.

Lemma 5.2. Let G be a compact semi-simple Lie group, K ⊂ G a compact subgroup,

and let M = G/K the naturally reductive homogeneous space equipped with the Rie-

mannian metric induced by −B. For every K-representation π on E, let EM := G×πE

be the associated vector bundle over M. Then the endomorphism q(¯R) acts fibre-wise

on EM as q(¯R) = −CasK

sections of EM, considered as G-representation via the left-regular representation, as

¯∆ = −CasG

Proof. Consider the Ad(K)-invariant decomposition g = k⊕p. For any vector X ∈ g we

write X = Xk+Xp, with Xk∈ k and Xp∈ p. The canonical homogeneous connection is

the left-invariant connection in the principal K-fibre bundle G → G/K corresponding

to the projection X ?→ Xk. It follows that one can do for the canonical homogeneous

connection on G/K the same identifications as for the Levi Civita connection on Rie-

mannian symmetric spaces.

π. Moreover the differential operator¯∆ acts on the space of

ℓ.

In particular, the covariant derivative of a section φ ∈ Γ(EM) with respect to

some X ∈ p translates into the derivative X(ˆφ) of the the corresponding function

ˆφ ∈ C∞(G;E)K, which is minus the differential of the left-regular representation X(ˆφ) =

−ℓ∗(X)ˆφ. Hence, if {eµ} is an orthonormal basis in p, the rough Laplacian¯∇∗¯∇ trans-

lates into the sum −ℓ∗(eµ)ℓ∗(eµ) = (−CasG

to show that q(¯R) = −CasK

We claim that the differential i∗ : k → so(p)∼= Λ2p of the isotropy representation

i : K → SO(p) is given by i∗(A) = −1

(1

Next we recall that for X,Y ∈ p the curvature¯RX,Y of the canonical connection acts by

−π∗([X,Y ]k) on every associated vector bundle EM, defined by the representation π.

ℓ+CasK

ℓ). Since¯∆ =¯∇∗¯∇+q(¯R) it remains

πin order to complete the proof of the lemma.

ℓ= −CasK

2eµ∧ [A,eµ] for any A ∈ k. Indeed

2B(eµ,X)[A,eµ] +1

2eµ∧ [A,eµ])∗X = −1

2B([A,eµ],X)eµ= −[A,X].

Page 14

14ANDREI MOROIANU AND UWE SEMMELMANN

Hence the curvature operator¯R can be written for any X,Y ∈ p as

¯R(X ∧ Y ) =1

Let PSO(p)= G ×iSO(p) be the bundle of orthonormal frames of M = G/K. Then any

SO(p)-representation ˜ π defines a K-representation by π = ˜ π ◦ i. Moreover any vector

bundle EM associated to PSO(p)via ˜ π can be written as a vector bundle associated via

π to the K-principle bundle G → G/K, i.e.

EM = PSO(p)×˜ πE = G ×πE

Let {fα} be an orthonormal basis of k. Then by the definition of q(¯R) we have

q(¯R) =

2eµ∧¯RX,Yeµ= −1

2eµ∧ [[X,Y ]k,eµ] = i∗([X,Y ]k).

1

2˜ π∗(eµ∧ eν) ˜ π∗(¯R(eµ∧ eν)) =1

= −1

=

2˜ π∗(eµ∧ eν)π∗([eµ,eν]k)

2B(eν,[fα,eµ])˜ π∗(eµ∧ eν)π∗(fα)

2B([eµ,eν],fα)˜ π∗(eµ∧ eν)π∗(fα) = −1

2˜ π∗(eµ∧ [fα,eµ])π∗(fα) = −π∗(fα)π∗(fα)

= −CasK

We have shown that q(¯R) ∈ End(EM) acts fibre-wise as −CasK

f ∈ C∞(G;E)K, then the K-invariance of f implies π∗(Z)f = −Z(f) = ℓ∗(Z)f and

also CasK

1

π.

π. Let Z ∈ k and

π= CasK

ℓ, which concludes the proof of the lemma.

?

It follows from this lemma that the spectrum of¯∆ on sections of EM is the set

of numbers λγ = ?ρ + γ?2− ?ρ?2, where γ is the highest weight of an irreducible G-

representation Vγsuch that HomK(Vγ,E) ?= 0, i.e. such that the decomposition of Vγ,

considered as K-representation, contains components of the K-representation E.

5.3. Nearly K¨ ahler deformations and Laplace eigenvalues. Let (M,g,J) be a

compact simply connected 6-dimensional nearly K¨ ahler manifold not isometric to the

round sphere, with scalar curvature normalized to scalg= 30. Recall the following result

from [10]:

Theorem 5.3. The Laplace operator ∆ coincides with the Hermitian Laplace operator

¯∆ on co-closed primitive (1,1)-forms. The space NK of infinitesimal deformations of

the nearly K¨ ahler structure of M is isomorphic to the eigenspace for the eigenvalue 12

of the restriction of ∆ (or¯∆) to the space of co-closed primitive (1,1)-forms on M.

Assume from now on that M is a 6-dimensional naturally reductive 3-symmetric space

G/K in the list of Gray and Wolf, i.e. SU2× SU2× SU2/SU2, SO5/U2or SU3/T2. As

was noticed before, the canonical homogeneous and the canonical Hermitian connection

coincide, since for the later can be shown that is torsion and its curvature are parallel,

a property, which by the Ambrose-Singer-Theorem characterizes the canonical homoge-

neous connection (c.f. [5]). In order to determine the space NK on M we thus need to

apply the previous calculations to compute the¯∆-eigenspace for the eigenvalue 12 on

primitive (1,1)-forms and decide which of these eigenforms are co-closed.

Page 15

THE HERMITIAN LAPLACE OPERATOR ON NEARLY K¨AHLER MANIFOLDS15

According to Lemma 5.2 and the decomposition (29) we have to carry out three

steps: first to determine the K-representation Λ1,1

to compute the Casimir eigenvalues with the Freudenthal formula, which gives all pos-

sible¯∆-eigenvalues and finally to check whether the G-representation Vγrealizing the

eigenvalue 12 satisfies HomK(Vγ,Λ1,1

0p defining the bundle Λ1,1

0TM, then

0p) ?= {0} and thus really appears as eigenspace.

Before going on, we make the following useful observation

Lemma 5.4. Let (G/K,g) be a 6-dimensional homogeneous strict nearly K¨ ahler man-

ifold of scalar curvature scalg= 30. Then the homogeneous metric g is induced from

−1

Proof. Let G/K be a 6-dimensional homogeneous strict nearly K¨ ahler manifold. Then

the metric is induced from a multiple of the Killing form, i.e. G/K is a normal homo-

geneous space with Ad(K)-invariant decomposition g = k ⊕ p. The scalar curvature of

the metric h induced by −B may be computed as (c.f. [3])

scalh=3

12B, where B is the Killing form of G.

2− 3CasK

λ

where λ : K → so(p) is the isotropy representation. From Lemma 5.2 we know that

CasK

2scalh

15

id. Hence we obtain the equation scalh=3

the metric g corresponding to −1

5.4. The¯∆-spectrum on S3× S3. Let K = SU2 with Lie algebra k = su2 and

G = K ×K ×K with Lie algebra g = k⊕k⊕k. We consider the 6-dimensional manifold

M = G/K, where K is diagonally embedded. The tangent space at o = eK can be

identified with

p = {(X,Y,Z) ∈ k ⊕ k ⊕ k|X + Y + Z = 0}.

Let B be the Killing form of k and define B0= −1

that the invariant scalar product

λ= −q(¯R), which on the tangent bundle was computed in Lemma 15 as q(¯R) =

2+2

5scalhand it follows scalh=5

2, i.e.

12B has scalar curvature scalg= 30.

?

12B. Then it follows from Lemma 5.4

B0((X,Y,Z),(X,Y,Z)) = B0(X,X) + B0(Y,Y ) + B0(Z,Z)

defines a normal metric, which is the homogeneous nearly K¨ ahler metric g of scalar

curvature scalg= 30.

The canonical almost complex structure on the 3-symmetric space M, corresponding

to the 3rd order G-automorphism σ, with σ(k1,k2,k3) = (k2,k3,k1), is defined as

J(X,Y,Z) =

2

√3(Z,X,Y ) +

1

√3(X,Y,Z).

The (1,0)-subspace p1,0of pCdefined by J is isomorphic to the complexified adjoint

representation of SU2on suC

(notice that E∼=¯E because every SU2∼= Sp1representation is quaternionic).

Lemma 5.5. The SU2-representation defining the bundle Λ(1,1)

ducible summands Sym4E and Sym2E.

2. Let E = C2denote the standard representation of SU2

0

TM splits into the irre-

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