Cayley submanifolds of Calabi-Yau 4-folds

0.54 MB·

by  Isabel M. C. Salavessa

Cayley submanifolds of Calabi-Yau 4-folds.
6
0 Isabel M.C. Salavessa1 and Ana Pereira do Vale2
∗ ∗∗
0
2
Dedicated to Jim Eells

n
a
J
1 Centro de F´ısica das Interac¸c˜oes Fundamentais, InstitutoSuperior T´ecnico, Edif´ıcio Ciˆencia,
0 Piso 3, 1049-001 LISBOA,Portugal; e-mail: [email protected]
3
2 Centro de Matem´atica, UniversidadedoMinho, Campus deGualtar

4710-057 BRAGA,Portugal; e-mail: [email protected]
]
G
D Abstract: Our main results are: (1) The complex and Lagrangian points of a non-complex and non-
. Lagrangian 2n-dimensional submanifold F :M N, immersed with parallel mean curvature and with
h
→
t equal K¨ahler angles into a K¨ahler-Einstein manifold (N,J,g) of complex dimension 2n, are zeros of
a
m finite order of sin2θ and cos2θ respectively, where θ is the common J-K¨ahler angle. (2) If M is a
[ CayleysubmanifoldofaCalabi-Yau(CY)manifoldN ofcomplexdimension4,then 2+NM isnaturally
isomorphic to 2 TM. (3) If N is Ricci-flat (not necessarily CY) and M is a Cayley submanifold, then
2 + V
v p1( 2+NM) =Vp1( 2+TM) still holds, but p1( 2−NM)−p1( 2−TM) may describe a residue on the
6 J-complex points, in the sense of Harvey and Lawson. We describe this residue by a PDE on a natural
0 V V V V
morphismΦ:TM NM,Φ(X)=(JX)⊥,withsingularitiesatthe complexpoints. Wegiveanexplicit
2 →
8 formula of this residue in a particularcase. When (N,I,J,K,g) is a hyper-K¨ahlermanifold and M is an
0 I-complexclosed4-submanifold,thefirstWeylcurvatureinvariantofM maybedescribedasaresidueon
4
the J-K¨ahler angle at the J-Lagrangianpoints by a Lelong-Poincar´etype formula. We study the almost
0
h/ complex structure Jω on M induced by F.
t
a
m 1 Introduction
:
v
Theroleofthecomplexandanti-complexpointsonthetopology-geometry ofclosednon-complex
i
X
minimal surfaces immersed into complex Ka¨hler surfaces has been studied in [28], [9], [8], and
ar [31]. In these papers, it is proved that the set = + − of complex and anti-complex points
C C ∪C
is a set of isolated points, and each of such points is of finite order. The order of the complex
and anti-complex points is defined as a multiplicity of a zero of (1 cosθ), where θ is the Ka¨hler
±
angle, and adjunction formulas were obtained in [28], [8] and [31]:
order(p) order(p)= (M)+ (NM) (1.1)
− − X X
p − p +
X∈C X∈C
order(p) order(p) = F c (N)[M]. (1.2)
∗ I
−
p − p +
X∈C X∈C
MSC 2000: Primary:53C42,53C55,53C25,53C38; Secondary:57R20,57R45.
Key Words: Minimal submanifold, K¨ahler angles, Cayley submanifolds, K¨ahler-Einstein manifold, Residue.
∗ and ∗∗ Partially supported by Funda¸c˜ao Ciˆencia e Tecnologia through POCI/MAT/60671/2004.
∗Partially supported by Funda¸c˜aoCiˆencia e Tecnologia through Plurianual of CFIF.
1
Salavessa–Pereira do Vale 2
The proofs of these formulas come, respectively, from the following PDEs of second order on the
cosine of the Ka¨hler angle, with singularities at complex and anti-complex points:
1∆logsin2θ = (KM +K⊥) (1.3)
2
1+cosθ
1∆log = RicciN(e ,e ), (1.4)
2 1 cosθ − 1 2
(cid:18) − (cid:19)
where KM and K are respectively the Gaussian curvature of M and the curvature of the
⊥
normal bundle NM, and e ,e is a direct orthonormal frame of M. Therefore, (1.1) and (1.2)
1 2
are formulas that describe some polynomials of topological invariants of the immersed surface,
normalbundleandambientspace,asresidueformulasofcertainfunctionsthathavesingularities
at those special points.
Inhigherdimensions,thepapers[17], [29],[30]showhowPontrjaginclasses andEulerclasses
of a closed (generic) submanifold M of a complex manifold (N,J) are carried by subsets of CR-
singular points, that is, points with sufficiently many complex directions. The investigation of
complex tangents on a m-dimensional submanifold M embedded into a Ka¨hler manifold N of
complex dimension m is very much justified, by the well known embedding theorem of Whit-
ney. More generally, if N has a calibration Ω of rank m (see definitions in [12]) and M is not
Ω-calibrated we may expect that Ω-calibrated points may have a similar role ([21]). Minimality
of M should guarantee the order of such points to be finite.
In [13], [14] a general framework is shown to obtain this sort of geometric residues, inspired
by the above examples. Given two Riemannian vector bundles (E,g ), (F,g ) over M, of the
E F
E F
same rank m, with Riemannian connections and , and a bundle map Φ : F E,
∇ ∇ →
degenerated at a set of points Σ, we may compare a m-characteristic classe Ch of E and the
F E
one of F, describing these invariants using the curvature tensors with respect to and ,
∇ ∇
via Chern-Weil theory. Φ induces on F a singular connection ′= Φ−1∗ E, Riemannian for
∇ ∇
a degenerated metric, and that makes Φ a parallel isometric bundle map, but R and Ch(R)
′ ′
can be smoothly extended to Σ by the identities R(X,Y,Z,W) = g (RE(X,Y)Φ(Z),Φ(W)),
′ E
Ch(R) = Ch(RE). ThedifferenceCh(R) Ch(RF)isoftheformdT whereT isatransgression
′ ′
−
form with singularities along Σ. If Σ is sufficiently small and regular, the Stokes theorem reads
dT = T, where V (Σ) is a tubular neighbourhood of Σ of radius ǫ, and letting
M Vǫ(Σ) − ∂Vǫ(Σ) ǫ
ǫ ∼ 0 may describe Ch(E) Ch(F) as a residue of T along Σ and expressed in terms of the
R → R −
zeros of Φ.
Inspired in this framework, the present paper shows some formulas of the type (1.3)-(1.4)
for 4-dimensional submanifolds of certain Ka¨hler manifolds. As we will see, to workout such
formulas in dimension > 2 is considerably more difficult then in the surface case.
We study the set of complex points and the set of Lagrangian points of non-holomorphic
C L
and non-Lagrangian immersed submanifolds F : M N of real dimension 2n of a Ka¨hler-
→
Einstein (KE) manifold of complex dimension 2n, namely if F is immersed with equal Ka¨hler
angles (e.k.a.s). A natural bundle map Φ : TM NM, Φ(X) = (JX)⊥, is defined and was
→
first used by Webster. Φ is degenerated at points with complex directions, and has maximum
norm at Lagrangian points, where it is an isometry. If F has e.k.a.s, Φ is conformal with
Salavessa–Pereira do Vale 3
Φ(X) 2 = sin2θ X 2 where θ is the common Ka¨hler angle, and away from , one can define
k k k k L
smooth almost complex structures J on M and J on the normal bundle NM that are natu-
ω ⊥
rally inherited from the ambient space, and they coincide with the induced complex structure at
complex points. These almost complex structures, with the Ka¨hler angle, will be fundamental
for our formulas. In section 3, if n = 2 we study J .
ω
If n = 2 and (N,J,g) is Ricci-flat KE, M is a Cayley submanifold if it is minimal and with
equal Ka¨hler angles. If N is Calabi-Yau these Cayley submanifolds are calibrated by one of
the S1-family of Cayley calibrations ([12],[16]). The Cayley calibrations Ω do not specify the
complex or the Lagrangian points, but induce a natural isomorphism ΩM : 2 TM 2 NM,
+ → +
ΩM(X Y),U V = Ω(X,Y,U,V) (see Prop.3.2)
h ∧ ∧ i V V
In Section 4 we prove that complex and Lagrangian points of a n-submanifold with parallel
mean curvature are zeros of a system of complex-valued functions that satisfy a second-order
partial differential system of inequalities of the Aronszajn type, and so, if the submanifold is
not complex or Lagrangian, they are zeros of finite order. These inequalities are obtained from
some estimates on the Laplacian of the pull-back of the Ka¨hler form of N by F, and on the
Laplacian of Φ : TM NM. Furthermore, the sets and have Hausdorff codimension at
→ C L
least 1, and if M is closed and n= 2, is a set of Hausdorff codimension at least 2.
L
In Section 5 we prove the following residue-type formula, in the same spirit as formulas (1.3)
and (1.4):
Theorem 1.1. If F : M N is a non-J-holomorphic Cayley submanifold immersed into a
→
4-fold Ricci-flat Ka¨hler manifold (not necessarily Calabi-Yau), the following equalities hold, for
some representatives in the cohomology classes of M:
p ( 2NM) = p ( 2TM) (1.5)
1 + 1 +
1
p (V2NM) = p (V2TM)+ dη (1.6)
1 − 1 − π2
V V
where η = η(Φ) is a 3-form, defined away from the complex points, which is given by
1 1
η(Φ) = Φ 1 Φ (Φ(R )+RM + [Φ 1 Φ,Φ 1 Φ]) (1.7)
− ⊥ − −
−4h ∇ ∧ 3 ∇ ∇ i
where Φ(R ) : 2TM 2TM is given by Φ(R )(X,Y)(Z) = Φ 1R (X,Y)Φ(Z). Further-
⊥ ⊥ − ⊥
→
more:
V V
(A) If F has no complex points TM and NM have the same Pontrjagin and Euler classes.
(B) If dΦ = 0, or if g( Φ(Y),Φ(Z)) is skew symmetric on (Y,Z), then θ is constant and
X
∇
Φ :TM NM is a parallel homothetic diffeomorphism.
→
(C) If g( Φ(Y),Φ(Z)) is symmetric on (Y,Z), or if R¯(X,Y)Φ = 0, where R¯ is the curvature
X
∇
tensor of TM NM, then p ( 2 TM) =p ( 2 NM) holds, or equivalently M and NM have
∗ 1 1
⊗
− −
the same Pontrjagin and Euler classes.
V V
We will say that F has regular homogeneous complex points , if = Σ is a disjoint finite
C i i
union of closed submanifolds Σ of dimension d 3, and for each i, on a neighbourhood V of
i i
≤ S
Σ in M, sinθ = fri where r is the common order of the zeros of Φ (and of Φ ) along Σ ,
i i i k k i
Salavessa–Pereira do Vale 4
and f is a nonnegative continuous function, smooth on V , such that f exists as a
i i
∼ C k∇ k
positive Cµ function on all V, with µ ≥ ri +2 and the flow of Xfi = ∇ffii2 can be extended
to G = (p,w) NΣ : w < t as a Cµ+1 diffeomorphism ξ : G k∇ kV. That is, X is
t0 { ∈ i k k 0} t0 → fi
a multivalued vector field at points p Σ , with sublimits spanning all T Σ and for each u
∈ i p ⊥i
unit vector of T Σ it is defined an integral curve γ (t) = ξ(p,tu) with γ (0) = p and
p ⊥i (p,u) (p,u)
γ (0) = u , where c(p) = f (p). This flow map ξ defines for each sufficiently small
(′p,u) c(p) k∇ ik
ǫ > 0 a diffeomorphism from C = (p,w) NΣ : w = ǫ onto f 1(ǫ). Furthermore, for
ǫ { ∈ i k k } i−
each sufficiently small coordinate chart y of Σ we have a Farmi-type coordinate chart x of V
of class Cµ+1, extending y and satisfying f = x2 +...+x2 (see Prop. 5.8). Examples of
i di+1 4
such functions f are the distance function σ toqa submanifold Σ . Let π : N1Σ := C Σ ,
i i i 1 i
→
π(p,u) = p, and S(p,1) the unit sphere of T Σ T M. For u S(p,1) and X T M, set
p ⊥i ⊂ p ∈ ∈ p
X u = X g(X,u)u, and define ς(u)(X) = ( X˜) T M where X˜ is any smooth section of
⊥ u ⊥ p
− ∇ ∈
TM with X˜ = X.
p
Corollary 1.1. Assume M is compact and F in Theorem 1.1 has regular homogeneous complex
points of order r on Σ . Let Φ˜ = Φ and set for each (p,u) N1Σ and X vector field on M,
i i Φ ∈ i
k k
Υ (p,u) := ri Φ(p), Ψ (p,u)(X ):= ri Φ(p),
i ∇uri i p ∇uri−1,X (1.8)
G (p,u)(X ):= (ri+1)Φ(p)+r Ψ (p,u)(ς(u)(X ))
i p ∇uri,X i i p
defining smooth sections Υ of π 1(TM NM) and Ψ ,G of π 1(TM (TM NM)).
i − ∗ i i − ∗ ∗
⊗ ⊗ ⊗
Then there exist the following limits
1
limΦ˜(ξ(p,ǫu)) = Υ (p,u), (1.9)
ǫ 0 ri!c(p)ri i
→
1
limǫ Φ˜(ξ(p,ǫu)) = Ψ (p,u)(X u) (1.10)
ǫ→0 ∇X (ri−1)!c(p)ri−1 i ⊥
1
if X f near p, lim Φ˜(ξ(p,ǫu)) = G (p,u)(X ), (1.11)
⊥∇ i ǫ 0∇X ri!c(p)ri i p
→
with 1 Υ (p,u) :T M NM an isometry. Furthermore, set
ri!c(p)ri i p → p
T(1)(p,u)(X) = Υ (p,u) 1 Ψ (p,u)(X) T(0)(p,u)(X) = Υ (p,u) 1 G (p,u)(X).
i i − ◦ i i i − ◦ i
Then
p ( 2NM)[M] p ( 2TM)[M] =
1 − − 1 −
V V
= ri ( c(p) 1 T(1)(p,u) (Υ(p,u)(R )+RM) ( u)d (u))d (p)
−4 − h i ∧ ⊥ i ∗ S(p,1) Σi
i:Xdi=2 ZΣi ZS(p,1)
+ 3 ri3−k ( c(p) 1 T(α)(p,u) [T(β)(p,u),T(γ)(p,u)] ( u)d (u))d (p).
− 12 − h i ∧ i i i ∗ S(p,1) Σi
Xk=0i:Xdi=k α+βX+γ=3−k ZΣi ZS(p,1)
In section 6 we prove that if M is a J-complex submanifold and N is Ricci-flat, then (1.5)
still holds. Moreover, if c (M) = 0, then 2 TM and 2 NM are both flat, and 2 TM and
1 + +
−
V V V
Salavessa–Pereira do Vale 5
2 NM are both anti-self-dual.
−
V If N = (N,I,J,K,g) is hyper-Ka¨hler (HK) of complex dimension 4, and M is an I-complex
submanifold of complex dimension 2, then, considering on N the complex structure J, M is a
Cayley submanifold with a J-Ka¨hler angle θ that can assume any value. Furthermore, M has a
hyper-Hermitian structure (M,I,J ,J ,g), defined away from totally complex points. More
ωJ ωK
generally, if M is an ”I-Ka¨hler” Cayley submanifold of a Ricci-flat Ka¨hler 4-fold (N,J,g), i.e.,
locallyonaopendensesetofM ,asmoothKa¨hlerstructureI existsandthatanti-commutes
∼ L
with J , then we conclude (in subsection 3.2) that the J-Ka¨hler angle θ also satisfies the PDE
ω
∆logcos2θ = sM (1.12)
where sM is the scalar curvature of M. If M is closed, this is a residue-type formula for the first
curvature invariant of Weyl of M, κ (M) = 1 sMVol , in terms of the zero set Σ of cosθ,
2 2 M M
which is the set of the the J-Lagrangian points of M. We prove in section 7:
RL
Proposition 1.1. If N is HK and M is a non-totally complex closed I-Ka¨hler submanifold
with I-Ka¨hler form ω , then there exist a locally finite union of irreducible analytic subvarieties
I
of complex codimension 1 (i.e analytic surfaces) Σ and integers a such that Σ = Σ where
i i i i
cosθ vanish at homogeneous order a along Σ and a formula of Lelong-Poincar´e type in terms
i i
S
of characteristic divisors exist: 1κ (M) = a ω .
π 2 − i i Σi I
P R
If I does not exist globally on M, we still can obtain a residue formula under some conditions,
and a removable high rank singularity theorem (see Proposition 7.1 and Corollary 7.1).
Insection8wegivesomeexamplesofcompletenon-linearCayleysubmanifoldsof(R8,J ,g ),
0 0
with no complex J -points, with only one complex point, with a 2-plane set of complex points,
0
or with a 2-plane set of Lagrangian points. They are all holomorphic for some other complex
structure of R8. We also observe that all submanifolds, and in particular coassociative ones,
that are graphs of maps f : R4 R3, do not have J -complex points.
0
→
2 The K¨ahler angles
We recall the notion of Ka¨hler angles introduced in [22], [23] and [26] for an immersed 2m-
submanifoldF :M N ofaKa¨hlermanifoldofcomplexdimension2nwherem n. Wedenote
→ ≤
by J and g the complex and Hermitian structureof N and ω(X,Y) = g(JX,Y) its Ka¨hler form.
Let NM = (dF(TM)) be the normal bundle and denote by ( ) the orthogonal projection of
⊥ ⊥
F 1TN onto NM. The pullback 2-form F ω defines at each point p M the Ka¨hler angles
− ∗
∈
θ ,...,θ of M, θ [0, π], such that cosθ ... cosθ 0 and icosθ are the
1 m α ∈ 2 1 ≥ ≥ m ≥ {± α}α=1,...,m
eigenvalues of the complex extension F ω to TcM. Polar decomposition of the endomorphism
∗ p
(F ω)♯ = (F ω)♯ J where ♯ is the usual musical isomorphism, defines a partial isometry J :
∗ ∗ ω ω
| |
m
TM TM with the same kernel as F ω. Then M = where a point p M is in
→ Kω ∗ k=0Lm−k ∈
iff Rank(F ω) = 2k. At each p , we may take an o.n. basis of T M of the form
m k ∗ p m k p
L − ∈ L − S
X ,Y = J X ,...,X ,Y = J X ,X ,Y ,...,X ,Y where X ,Y ,...,X ,Y
1 1 ω 1 k k ω k k+1 k+1 m m k+1 k+1 m m
{ } { }
is any o.n. basis of . If O is an open set of M lying in , then X ,Y can be chosen
ω m k α α
K L −
Salavessa–Pereira do Vale 6
smoothly on a neighbourhood of each point of O. The complex frame
X iY
α = Z := α− α α¯ = Z¯ α = 1,...,m (2.1)
α α
2
diagonalizes F ω, (F ω)♯(Z )= icosθ Z , and for α k, Z T(1,0)M w.r.t. J . We will use
∗ ∗ α α α α ω
≤ ∈
the greek letters α,β,µ,... and their conjugates to denote both the integer in 1,2,...,m it
{ }
representsorthecorrespondingcomplex vector ofTcM abovedefinedin(2.1). IfM isorientable
then + and , are the set of points such that cosθ (p) = 1 α, and respectively, J defines
− α ω
C C ∀
the same or the opposite orientation of M. The eigenvalues cosθ are only locally Lipschitz on
α
M, while the product 2ǫ(p)cosθ ...cosθ = F ωm,Vol is smooth everywhere, where ǫ(p)
1 m ∗ M
h i
is the orientation of J for p . For E subspace of T M set EJ = E JE.
ω 0 p
∈ L ∩
Let ω = ω be the restriction of the Ka¨hler form ω of N to the normal bundle NM, and
⊥ NM
(ω )♯ = (ω| )♯ J be its polar decomposition. We define the following morphisms
⊥ ⊥ ⊥
| |
Φ : TM NM Ξ: NM TM
→ →
X (JX) U (JU)
⊥ ⊤
→ →
Note that JX = (F ω)♯(X)+Φ(X), and Φ(X) = 0 iff X,JX is a complex direction of F.
∗
{ }
Similarlyforω andΞ. Φ : (TMJ) NM,Ξ : (NMJ) TM are1-1. Set2s = dim(TMJ),
⊥ ⊥ ⊥
→ →
2t =dim(NMJ). Theo.n.basis U ,V = U ,JU ,...,U ,JU ,Φ( Yα ), Φ( Xα ) , whereα
{ A A} { 1 1 t t sinθα sinθα }
ares.t. sinθ =0, diagonalize ω , andso2n = 2m+t s,andfor A= α+t s,σ = θ arethe
α ⊥ A α
6 − −
thenon-zeroKa¨hlerangles of NM. Thatis,TM andNM have thesamenonzeroKa¨hlerangles,
and they have the same multiplicity. Only the eigenvalues i of (F ω)♯ and of ω may or not
∗ ⊥
±
exist and may appear with different multiplicity, t and s, respectively. Set E = span X ,Y ,
α α α
{ }
F = span U ,V , andP , P the correspondingorthonormal projections ofTM and NM.
A { A A} Eα FA
We use the Hilbert-Schmidt inner products on tensors and forms. We have
kF∗ωk2 = 12k(F∗ω)♯k2 = αcos2θα = kω⊥k2+(s−t)= kω⊥k2−2(n−m)
g(Φ(X),Φ(Y)) = (1 cosθ cosθ )g(X,Y) for X E ,Y E (2.2)
− Pα β ∈ α ∈ β
g(Ξ(U),Ξ(V)) = (1 cosθ cosθ )g(U,V) for U F ,V F
α β α β
− ∈ ∈
Φ 2 = 2 sin2θ = Ξ 2 4(n m)
k k α α k k − −
ω Φ = Φ (F ω)♯ (F ω)♯ Ξ = Ξ ω
⊥◦ P− ◦ ∗ ∗ ◦ − ◦ ⊥
J Φ = Φ J J Ξ = Ξ J on (2.3)
⊥ ω ω ⊥ 0
◦ − ◦ ◦ − ◦ L
Ξ Φ = sin2θ P Φ Ξ= sin2θ P . (2.4)
− ◦ α α Eα − ◦ α α Fα
P P
If X T M and U NM , then ω(U,Φ(X)) = ω(Ξ(U),X), ω(U,J X) = ω(J U,X),
p p ω ⊥
∈ ∈
g(U,Φ(X) ) = g(Ξ(U),X).
i −
We denote by bothLevi-Civita connections of M and N or F 1TN, if noconfusion exists,
−
∇
M N
otherwise we explicit them by and . We take on NM the usual connection ⊥, given
∇ ∇ ∇
by X⊥U = ( XU)⊥, for X and U smooth sections of TM and NM F−1TN, respectively.
∇ ∇ ⊂
We denote the corresponding curvature tensors by RM, RN and R . The sign convention we
⊥
choose for the curvature tensors is R(X,Y)Z = Z + + Z. The second
X Y Y Z [X,Y]
−∇ ∇ ∇ ∇ ∇
fundamental form of F, dF(Y) = dF(X,Y) is a symmetric 2-tensor on M that takes
X
∇ ∇
values on the normal bundle. Its covariant derivative ⊥ dF is defined considering dF with
∇ ∇ ∇
Salavessa–Pereira do Vale 7
values on NM. We denoteby i : NM F 1TN theinclusion bundlemap, andits covariant
NM → −
derivative i is a morphism from NM into TM. Then X,Y T M, U NM , p M,
∇X NM ∀ ∈ p ∈ p ∈
g( i (U),Y)=g(( NU)⊤,Y)= g(U, dF(Y))= g(AU(X),Y)= g(U,( NY)⊥) (2.5)
∇X NM ∇X − ∇X − − ∇X
whereA :NM L(T M;T M)istheshapeoperator. LetH = 1 trace dF denotethe
p → p p dim(M) gM∇
mean curvatureof F. F is minimal (resp. with parallel mean curvature) if H = 0 (resp. ⊥H =
∇
0). F is J -pluriminimal in if ( dF)(1,1)(X,Y)= 1( dF(X,Y)+ dF(J X,J Y)) = 0. In
ω L0 ∇ 2 ∇ ∇ ω ω
this case F is minimal on . For p M, X,Y,Z T M, U,V NM ,
0 p p
L ∈ ∈ ∈
F ω(X,Y) = g( dF(X),Φ(Y))+g( dF(Y),Φ(X)) (2.6)
Z ∗ Z Z
∇ − ∇ ∇
ω (U,V) = g( i (U),Ξ(V))+g( i (V),Ξ(U)). (2.7)
∇Z ⊥ − ∇Z NM ∇Z NM
If (E,g ) is a Riemannian vector bundleand T,S :TM E are vector bundlemaps, we define
E
→
a 2-form T S by
h ∧ i
T S (X,Y) = g (T(X),S(Y)) g (T(Y),S(X)).
E E
h ∧ i −
From the symmetry of dF, dF dF (X,Y)= dF dF (Z,W). Recall the
Z W X Y
∇ h∇ ∧∇ i h∇ ∧∇ i
Gauss, Ricci and Coddazzi equations: For X,Y,Z C (TM), and U,V C (NM)
∞ ∞
∈ ∈
RM(X,Y,Z,W) = RN(X,Y,Z,W)+ dF dF (X,Y) (2.8)
Z W
h∇ ∧∇ i
R (X,Y,U,V) = RN(X,Y,U,V)+ AU AV (X,Y) (2.9)
⊥
h ∧ i
RN(X,Y,Z,U) = g( X⊥ dF(Y,Z) Y⊥ dF(X,Z) , U ). (2.10)
− ∇ ∇ −∇ ∇
2.1 ∆Φ,∆F ω
∗
Lemma 2.1. Let F : M N be a 2m-dimensional immersed submanifold. For any X,Y
→ ∈
T M, U,V NM , and any local o.n. frame e of M,
p p i
∈
(i) Φ(Y) = ω ( dF(Y)) dF((F ω)♯(Y)).
X ⊥ X X ∗
∇ ∇ −∇
(ii) dΦ(X,Y)= dF((F ω)♯(Y))+ dF((F ω)♯(X)).
X ∗ Y ∗
−∇ ∇
(iii) δΦ = 2m(JH) .
⊥
−
(iv) ∆Φ(X) =2m(∇(⊥F∗ω)♯(X)H −∇X⊥ω⊥(H)−ω⊥(∇X⊥H))+∇XdF(δ(F∗ω)♯)+
+ dF( (F ω)♯(X),e )+ (RN(e ,X,(F ω)♯(e )) RN(e ,(F ω)♯(X),e ))⊥.
i∇ ∇ei ∗ i i i ∗ i − i ∗ i
(v) ∆F∗Pω(X,Y) = 2m(g( X⊥H,ΦP(Y)) g( Y⊥H,Φ(X)))+2mg(H,dΦ(X,Y))
∇ − ∇
+Trace RN(X,Y,dF(),Φ())+ dF, Φ dF, Φ .
M X Y Y X
· · h∇ ∇ i−h∇ ∇ i
(vi) g(∇XΞ(U),Y) = g((F∗ω)♯(∇XiNM(U))−∇XiNM(ω⊥(U)),Y)= −g(∇XΦ(Y),U).
Proof. We take smooth vector fields X,Y of M such that at a given point p , Y(p ) =
0 0
∇
X(p ) = 0, and assume also that e = 0 at p . Then at p
0 i 0 0
∇ ∇
XΦ(Y) = X⊥(Φ(Y)) =( X(JdF(Y))⊥)⊥ = ( X(JdF(Y) (F∗ω)♯(Y)))⊥
∇ ∇ ∇ ∇ −
= (J dF(Y))⊥ dF((F ω)♯(Y)) = ω ( dF(Y)) dF((F ω)♯(Y))
X X ∗ ⊥ X X ∗
∇ −∇ ∇ −∇
Salavessa–Pereira do Vale 8
and we get (i). (ii) follows from dΦ(X,Y) = Φ(Y) Φ(X), and the symmetry of
X Y
∇ − ∇
dF. It follows that δΦ = Φ(e ) = 2m(JH) + dF((F ω)♯(e )). Note that
∇ − i∇ei i − ⊥ i∇ei ∗ i
dF((F ω)♯(e )) = 0 because dF(X,Y) is symmetric and (F ω)♯ is skew symmetric.
i∇ei ∗ i P ∇ P ∗
Then (iii) is proved. Now,
P
dδΦ(X) = 2md((JH)⊥)(X) = 2m X⊥(ω⊥(H)) = 2m X⊥ω⊥(H) 2mω⊥( X⊥H).
− − ∇ − ∇ − ∇
δdΦ(X) = − i∇eidΦ(ei,X) = − i∇e⊥i (dΦ(ei,X))
= −Pi∇e⊥i (−∇eidF((F∗Pω)♯(X))+∇XdF((F∗ω)♯(ei)))
= Pi∇e⊥i (∇(F∗ω)♯(X)dF(ei)−∇XdF((F∗ω)♯(ei)))
= P ∇e⊥i ∇dF((F∗ω)♯(X),ei)+∇dF(∇ei(F∗ω)♯(X),ei)
Xi −∇e⊥i ∇dF(X,(F∗ω)♯(ei))−∇XdF(∇ei((F∗ω)♯)(ei))
= ∇(⊥F∗ω)♯(X)∇dF(ei,ei)−(RN(ei,(F∗ω)♯(X))ei)⊥ −∇X⊥∇dF(ei,(F∗ω)♯(ei))
i
X +(RN(ei,X)(F∗ω)♯(ei))⊥ +∇dF(∇ei(F∗ω)♯(X),ei)+∇XdF(δ((F∗ω)♯))
where we applied Coddazzi’s equation (2.10) in the last equality. Since X⊥ dF is symmetric
∇ ∇
and F∗ω is skew symmetric i∇X⊥∇dF(ei,(F∗ω)♯(ei)) = 0. Thus,
P
δdΦ(X) = ∇(⊥F∗ω)♯(X)(2mH)+ i(RN(ei,X)(F∗ω)♯(ei)−RN(ei,(F∗ω)♯(X))ei)⊥
+ i∇dF(∇ei(F∗ω)♯P(X),ei)+∇XdF(δ((F∗ω)♯)).
P
From ∆Φ = (dδ + δd)Φ(X), we get the expression in (iv). Since F ω is closed, then, for Y
∗
vector field with Y(p ) = 0, and using (2.10) and (2.6)
0
∇
∆F ω(X,Y)= (dδ+δd)F ω(X,Y)= d(δF ω)(X,Y)=
∗ ∗ ∗
= (δF ω)(Y) (δF ω)(X) = d( F ω(e ,Y))(X)+d( F ω(e ,X))(Y)
∇X ∗ −∇Y ∗ i −∇ei ∗ i ∇ei ∗ i
( )
= d g( dF(e ,e ),Φ(Y)) g( dFP(e ,Y),Φ(e )) (X)
i ∇ i i − ∇ i i
( )
+ d g( dF(e ,e ),Φ(X))+g( dF(e ,X),Φ(e )) (X)
P i − ∇ i i ∇ i i
= d(Pg(2mH,Φ(Y)))(X)−d(g(2mH,Φ(X)))(Y)− ig(∇X⊥∇dF(ei,Y),Φ(ei))
− ig(∇dF(ei,Y),∇XΦ(ei))+g(∇Y⊥∇dF(ei,X)P,Φ(ei))+g(∇dF(ei,X),∇YΦ(ei))
⊥ ⊥
= 2mP(g( H,Φ(Y)) g( H,Φ(X)))+2mg(H, Φ(Y)) 2mg(H, Φ(X))
∇X − ∇Y ∇X − ∇Y
+ i g(−∇X⊥∇dF(Y,ei)+∇Y⊥∇dF(X,ei) , Φ(ei))
+ g( dF(e ,Y), Φ(e ))+g( dF(e ,X), Φ(e ))
Pi− ∇ i ∇X i ∇ i ∇Y i
= 2mP(g( ⊥H,Φ(Y)) g( ⊥H,Φ(X))+g(H,dΦ(X,Y)))+ RN(X,Y,e ,Φ(e )
∇X − ∇Y i i i i
+ g( dF(e ,Y), Φ(e ))+g( dF(e ,X), Φ(e ))
i− ∇ i ∇X i ∇ i ∇Y Pi
P
obtaining the expression ∆F ω of (v). Finally we prove (vi). From Ξ(U)+ω (U) = JU, and
∗ ⊥
assuming at a point p0 ∇⊥U(p0)= 0 ( and so ∇XNU = ∇XiNM(U)) we obtain, at p0
∇XΞ(U)+∇XiNM(ω⊥(U)) = (∇XN(Ξ(U)+ω⊥(U)))⊤ = (J∇XNU)⊤
= (J(∇XiNM(U)))⊤ = (F∗ω)♯(∇XiNM(U)).
Salavessa–Pereira do Vale 9
Therefore, using (2.5), and (i),
g( Ξ(U),Y) = g( i (U),(F ω)♯(Y))+g(ω (U), dF(Y))
∇X − ∇X NM ∗ ⊥ ∇X
= g( dF((F ω)♯(Y)),U) g(ω ( dF(Y)),U) = g( Φ(Y),U).
X ∗ ⊥ X X
∇ − ∇ − ∇
3 Cayley submanifolds
We assume m = n and that F : M2n N2n has equal Ka¨hler angles (e.k.a.s), that is θα = θ
→
α and we denote by = . In this case Φ and Ξ are conformal bundle maps and
n
∀ L L
(F ω)♯ = cosθJ ω =cosθJ (3.1)
∗ ω ⊥ ⊥
Ξ Φ = sin2θId Φ Ξ = sin2θId . on M. (3.2)
TM NM
− ◦ − ◦
On M , J is g -orthogonal. Thus α,β,
ω M
∼ L ∀
g( J (α),β) = 2ig( α,β) = g(α, J (β)), g( J (α),β¯)= 0. (3.3)
Z ω Z Z ω Z ω
∇ ∇ − ∇ ∇
The Ricci tensor of N can be expressed in terms of the frame (2.1) as (see [23])
sin2θRicciN(U,V) = 4RN(U,JV,α,Φ(α¯)) = Trace RN(U,JV,dF(),Φ()) (3.4)
α M · ·
P
valid at all points p M, and U,V T N. We have
F(p)
∈ ∈
Proposition 3.1. Assume (N,J,g) is KE with Ricci = Rg, and F : M N is a 2n-
→
dimensional immersed submanifold with e.k.a.s. Then
(1) dΦ(X,Y) = 2cosθ(∇dF)(1,1)(JωX,Y) and TraceC,JωdΦ = αdΦ(Xα,Yα) = 2ncosθ H.
(2) dsin2θ(X) g(Y,Z) = g( XΦ(Y),Φ(Z))+g( XΦ(Z),Φ(Y))P.
· ∇ ∇
(3) dΦ = 0 iff dΦ(X,J X) = 0 iff F is J -pluriminimal or Lagrangian. Furthermore: (a) if
ω ω
R = 0, then dΦ = 0 iff F is complex or Lagrangian; (b) if R = 0, dΦ = 0 iff F has constant
6
Ka¨hler angle and Φ : (TM, ,gM) (NM, ⊥,g) is a parallel homothetic morphism.
∇ → ∇
(4) δΦ = 0 iff H is a Lagrangian direction of NM, iff F is minimal away from . Conse-
L
quently, Φ : (TM, ,gM) (NM, ⊥) is closed and co-closed 1-form (and so harmonic) iff
∇ → ∇
Φ :(TM, ,gM) (NM, ⊥) is parallel iff F is Lagrangian or Jω- pluriminimal.
∇ → ∇
(5) If F has parallel mean curvature then
∆Φ(X) = −2n∇X⊥ω⊥(H)+∇XdF(δ(F∗ω)♯)+ i∇dF(∇ei(F∗ω)♯(X),ei) (3.5)
+ i(RN(ei,X,(F∗ω)♯(ei))−RN(ei,P(F∗ω)♯(X),ei))⊥ (3.6)
∆F ω(X,Y) = 2nPg( H, dF((F ω)♯(Y))+ dF((F ω)♯(X)) )
∗ X ∗ Y ∗
−∇ ∇
+sin2θRF ω(X,Y)+ 2ω ( dF(Y), dF(X))
∗ i ⊥ ∇ei ∇ei
+ YdF, XdF (F∗ωP)♯ XdF, YdF (F∗ω)♯ .
h∇ ∇ ◦ i−h∇ ∇ ◦ i
Proof. (1) follows from Lemma 2.1, (2) from differentiation of (2.2), (3) and (4) are consequence
of [24] and Lemma 2.1 (ii) and (iii), and (5) follows directly from Lemma 2.1., (3.4) and the
Salavessa–Pereira do Vale 10
J-invariance of RicciN.
Now we assume n = 2. Four dimensional submanifolds of any Ka¨hler manifold of complex
dimension 4, immersed with equal Ka¨hler angles, are justthe same as submanifolds satisfying
≥
F ω = F ω.
∗ ∗
∗ ±
Since pointwise F ω is self-dual or anti-self-dual, and is a closed 2-form, then it is co-closed as
∗
well. In particular it is an harmonic 2-form. This is not the case of n = 2, unless if θ = constant
6
(see [24], or nextlemma 3.1(2)). Incase that N is a KE manifold of zero Ricci tensor and of real
dimension 8, a Cayley submanifold is a minimal 4-dimensional submanifold with equal Ka¨hler
angles θ = θ = θ. If N is a Calabi-Yau 4-fold, that is a Ka¨hler manifold with a complex
1 2
(4,0)
volume form ρ C M (this condition implies Ricci-flat, and the converse also holds in case
∈
N is simply connected), these submanifolds are characterised by being calibrated by one of the
V
Cayley calibrations Ω = 1ω ω+Re(ρ), where ρ is one of the S1-family of parallel holomorphic
2 ∧
volumes of N ([12]). Calabi-Yau 4-folds are Spin(7) manifolds. So, locally on N there is a
section e ,...,e of the principal Spin(7)-bundle of frames of N defined on a open set U of
1 8
N, and{that at eac}h point p U, defines a isometry of T N onto R8 such that Ω looks like (see
p
∈
[16])
Ω = dx +dx +(dx +dx ) (dx +dx )
1234 5678 12 34 ∧ 56 78 (3.7)
+(dx dx ) (dx dx ) (dx +dx ) (dx +dx ).
13 24 57 68 14 23 58 67
− ∧ − − ∧
From this equation we see that the subspaces spanned by e ,...,e and e ,...,e are Cayley
1 4 5 8
subspaces. We note that we use the opposite orientation on Cayley subspaces that Harvey and
Lawson do in [12] , and the calibration they use is given by Ω = 1ω2+Re(ρ) that is
′ −2
Ω = dx +dx +(dx dx ) (dx dx )
′ 1234 5678 12− 34 ∧ 56− 78 (3.8)
+(dx +dx ) (dx +dx )+(dx dx ) (dx dx )
13 24 57 68 14 23 58 67
∧ − ∧ −
and so Ω and Ω differ on the chosen parallel holomorphic volume, giving opposite fase on
′
−
the special Lagrangian calibration. In [12] it is proved that Spin(7) acts transitively on the
grassmannian G(Ω) of Cayley 4-planes of R8 and the isotropic subgroup of a Cayley subspace
Z
E is K SU(2) SU(2) SU(2)/ (see in (1.39) of [12] how K is embedded in Spin(7)).
2
≡ × ×
Thus, we can assume that B = e ,e ,e ,e and B = e ,e ,e ,e are direct o.n. basis of
1 2 3 4 ⊥ 5 6 7 8
{ } { }
T M and NM respectively. We identify isometrically in the usual way bivectors with 2-forms.
p p
So JB = e e +e e , JB =e e e e , and JB = e e +e e defines a direct o.n.
1 1∧ 2 3∧ 4 2 1∧ 3− 2∧ 4 3 1∧ 4 2∧ 3
basis (of norm √2) of 2 T M. Similar for 2 NM . We define a bilinear map:
+ p + p
V Ω : 2T VM 2NM R
△ p p
× →
Ω (X Y,U V) = Ω(X,Y,U,V)
△ V∧ ∧ V
Proposition 3.2. Ω defines a natural orientation reversing isometric bundle isomorphism
△
between 2TM and 2NM.
+ +
Proof. IdVentifying isoVmetrically in the canonic way (via musical isomorphisms with respect to
the induced metrics) the bilinear map Ω restricted to 2 T M 2NM with a linear map
△ + p × + p
Ω : 2 T M 2 NM , and using the frame e adapted to M, from (3.7) we see that Ω
△ + p → + p i V V △
V V