COMPLEMENT TO: HAMILTONIAN STATIONARY TORI IN THE COMPLEX PROJECTIVE PLANE

DOI: 10.1112/S0000000000000000

COMPLEMENT TO: HAMILTONIAN STATIONARY TORI IN THE COMPLEX PROJECTIVE PLANE

FR´ ED´ ERIC H´ ELEIN and PASCAL ROMON

We present here an extended version of paragraphs 2.1. and 2.2 of [1].

0.1. The space C P ⁿ

The complex projective space C P ⁿ can be identified with the quotient manifold C ⁿ⁺¹ \ {0}

/ C ^⋆ . We denote by π : C ⁿ⁺¹ \ {0} −→ C P ⁿ the canonical projection (a.k.a. Hopf fibration): if z = (z ¹ , · · · , z ⁿ⁺¹ ) ∈ C ⁿ⁺¹ \ {0}, π(z) = [z] is the equiva- lence class of z modulo C ^⋆ (the complex punctured line spanned by z) and is also written in homogeneous coordinates [z ¹ : · · · : z ⁿ⁺¹ ].

The tangent bundle — If [z] ∈ C P ⁿ a tangent vector ξ ∈ T [z] C P ⁿ can be represented by a C -linear map ℓ : C z −→ C ⁿ⁺¹ , λz 7−→ ℓ(λz) such that

∀λ ∈ C , dπ λz (ℓ(λz)) = ξ. (0.1)

Note that ℓ is not unique and given some ξ ∈ T [z] C P ⁿ if ℓ and ˜ ℓ satisfy condition (0.1) with ξ, then there exists k ∈ C s.t. ˜ ℓ(λz) = ℓ(λz) + kλz. However by using the standard Hermitian product h·, ·i

on C ⁿ⁺¹ we can select an unique ℓ ⁰ which satisfies (0.1) and such that

hℓ ⁰ (z), zi

= 0. (0.2)

Indeed it suffices to start from any ℓ satisfying (0.1) and to set ℓ ⁰ (z) = ℓ(z) + kz.

Then condition (0.2) holds if and only if k = −

.

The Hermitian metric — We can define a Hermitian product on C P ⁿ as follows.

If ξ 1 , ξ 2 ∈ T [z] C P ⁿ we consider the linear maps ℓ ⁰ ₁ and ℓ ⁰ ₂ satisfying (0.2) and such that dπ z (ℓ ⁰ ₁ (z)) = ξ 1 and dπ z (ℓ ⁰ ₂ (z)) = ξ 2 . Then the Hermitian product of ξ 1 and ξ 2 is

hξ 1 , ξ 2 i

:= hℓ ⁰ ₁ (z), ℓ ⁰ ₂ (z)i

|z| ²

,

a definition which is obviously invariant by transformations z 7−→ λz. If we had started with linear mappings ℓ 1 , ℓ 2 which lift respectively ξ 1 and ξ 2 according to (0.1) but without the condition (0.2) we could recover hξ 1 , ξ 2 i

by substitution of ℓ ⁰ a (z) = ℓ a (z) −

^(z),zi

Cn+1

z (for a = 1, 2) in the above definition. It gives hξ 1 , ξ 2 i

= hℓ 1 (z), ℓ 2 (z)i

|z| ²

− hℓ 1 (z), zi

hz, ℓ 2 (z)i

|z| ⁴

. (0.3)

2000 Mathematics Subject Classification 53C55 (primary), 53C42, 53C25, 58E12 (secondary).

Note that the Hermitian metric h·, ·i

on C P ⁿ provides us with an Euclidean metric h·, ·i E and a symplectic form ω through

h·, ·i

= h·, ·i E − iω(·, ·).

The horizontal distribution and the connection — For each z ∈ C ⁿ⁺¹ \ {0} we let H z to be the complex n-subspace in C ⁿ⁺¹ which is Hermitian orthogonal to z (and hence to the fiber of π). Note that the definition of the Hermitian metric on C P ⁿ is such that

¹

Cn+1

dπ z : H z −→ T [z] C P ⁿ is an isometry between complex Hermitian spaces (and dπ z allows us also to orient H z ). We call the subbundle H := ∪ z∈C

H z of T C ⁿ⁺¹ \ {0} −→ C ⁿ⁺¹ \ {0} the horizontal distribution.

It defines in a natural way a connection ∇

≃ ∇ ^H on the Hopf bundle π : C ⁿ⁺¹ \ {0} −→ C P ⁿ : if f : C P ⁿ −→ C ⁿ⁺¹ \ {0} is a section of this bundle its covariant derivative is

∇ ^H ξ f

[z] := hdf [z] (ξ), f(z)i

|f (z)| ²

f (z).

In order to compute the curvature of this connection, let us choose two vector fields ξ 1 and ξ 2 on C P ⁿ such that [ξ 1 , ξ 2 ] = 0. Then using (0.3) one computes that

∇ ^H _ξ

∇ ^H _ξ

f

− ∇ ^H _ξ

∇ ^H _ξ

f

= (hξ 2 , ξ 1 i

− hξ 1 , ξ 2 i

) f = 2iω(ξ 1 , ξ 2 )f.

Hence the curvature of this connection is proportional to the symplectic form ω. It has the following consequence:

Proposition 0.1. Let Ω be a simply connected open subset of R ⁿ and u : Ω −→

C P ⁿ be a smooth Lagrangian immersion, i.e. such that u

ω = 0. Then there exists a lift

C ⁿ⁺¹

π

Ω

b

u

< <

z z z z z z z z

u / / C P ⁿ

such that u

∇ ^H

u b = 0 (where u

∇ ^H is the pull-back by u of the connection ∇ ^H ).

Moreover this lift is unique modulo a multiplicative constant in C ^⋆ , is isotropic (i.e. the pull-back by b u of the symplectic form on C ⁿ⁺¹ vanishes) and we can choose it in such a way that | b u|

= 1 everywhere (i.e. such that u b : Ω −→ S ²ⁿ⁺¹ ).

Proof. The curvature of u

∇ ^H is the pull-back by u of the curvature of ∇ ^H , i.e. 2iu

ω. But this 2-form vanishes because u is Lagrangian and thus u

∇ ^H is flat. Hence we can find a parallel section b u (unique up to multiplication by a non- zero constant). The condition u

∇ ^H

u b = 0 writes hd b u, ui b

= 0, which implies d| b u| ²

= hd b u, ui b

+ h u, d b ui b

= 0. Hence | b u|

is constant and in partic- ular we can choose u b to take values in S ²ⁿ⁺¹ .

Observe that in Proposition 0.1 the tangent n-subspace along u b at (x, y) is a sub-

†

note the the sign convention may vary in the literature, e.g. some Authors use h·, ·i

=

h·, ·i

+ iω(·, ·).

space of H u(x,y)

: we then say that b u is a Legendrian immersion.

The Levi–Civita connection on C P ⁿ — It can be expressed in terms of the trivial connection D of ( C ⁿ⁺¹ , h·, ·i

) as follows. Let U ⊂ C P ⁿ be an open subset such that we can define a smooth local section σ : U −→ C ⁿ⁺¹ \ {0} of the Hopf bundle.

For any m ∈ U we denote by m b := σ(m). Let X be a smooth tangent vector field on C P ⁿ defined on U and, choosing some point m ∈ U , let ξ ∈ T m C P ⁿ . In order to define the covariant derivative of X at m along ξ we let ξ b := dσ m (ξ) ∈ T m

C P ⁿ and X b := σ

X . Then we set

(∇ ξ X ) (m) := dπ m

D ξ X b

(m)

− h ξ, b mi b

| m| b ²

X (m) + h X b (m), mi b

| m| b ²

ξ

!

(0.4) (Of course this definition does not depend on the choice of the local section σ.) One can then check easily that this connection respects the metric and is torsion-free and so it is the Levi–Civita connection on C P ⁿ .

0.2. The mean curvature vector of a Lagrangian submanifold in C P ⁿ

We consider here a simply connected Lagrangian submanifold Σ of C P ⁿ and denote by u : Σ −→ C P ⁿ its immersion map. According to Proposition 0.1, we can construct a Legendrian lift u b : Σ −→ C ⁿ⁺¹ \ {0}. We also consider a smooth orthonormal moving frame along u, i.e. sections E 1 , · · · , E n of u

T C P ⁿ such that for all x ∈ Σ, (E 1 (x), · · · , E n (x)) is a direct orthonormal basis over R of T u(x) Σ and, for all a, we let e a := u b

E a . Then for all x ∈ Σ, (e 1 (x), · · · , e n (x)) is a direct orthonormal basis over R of T u(x)

u(Σ). We next define the b mean curvature vector H ~ of the immersion u which is the normal component of the trace of the second funda- mental form. The main point here is that, since T u(x) Σ is Lagrangian, it is mapped by the complex structure

J of T u(x) C P ⁿ to the normal space T u(x) C P ⁿ

. And so (JE 1 (x), · · · , JE n (x)) is an orthonormal basis of the normal subspace to T u(x) Σ in T u(x) C P ⁿ . Hence

H ~ := 1 n

X n

a,b=1

h∇ E

E a , JE b i _E JE b . (0.5) Note that then, since he a , b ui

= he b , b ui

= 0, we have according to (0.4)

∇ E

E b = dπ u(x)

(D E

e b ) = dπ

u(x) (D e

e b ) . Hence

H ~ = 1 n

X n

a,b=1

hD e

e a , ie b i

JE b . (0.6) The complex volume n-form along b u — Let us consider on C ⁿ⁺¹ the complex volume (n + 1)-form Θ := dz ¹ ∧ · · · ∧ dz ⁿ⁺¹ . Then we construct the complex volume n-form along b u to be the section b θ of b u

(Λ ⁿ H

⊗ C ) by

θ b x := ( b u(x) Θ)

b

u(x)

, ∀x ∈ Σ,

†

the complex structure J is the image of the canonical complex structure i on H

by dπ

.

where is the interior product. Lastly we define the complex volume n-form θ along u to be the section of u

(Λ ⁿ T

C P ⁿ ⊗ C ) defined by θ := u b

θ. Using b θ b we define the Lagrangian angle function along u b to be the function β u

: Σ −→ R /2π Z such that

e ^iβ

^(x) = θ x (E 1 (x), · · · , E n (x)) = Θ( u(x), e b 1 (x), · · · , e n (x)), ∀x ∈ Σ.

It is not difficult to check that this definition is independent of the choice of the framing (E 1 , · · · , E n ). However all this construction depends on the choice of the Legendrian lift b u: another choice b v would lead to another complex volume n-form along ˆ v, and another Lagrangian angle function β

v , however β

u − β

v is a constant in R /2π Z .

Lemma 0.2. The mean curvature vector of the Lagrangian surface Σ can be computed using the Lagrangian angle function β

u through the relation

H ~ = 1

n J∇β u

. (0.7)

Proof. We set here β = β u

. Relation (0.7) is equivalent to proving that

∀V ∈ T x C P ⁿ , nh H, JV ~ i E = h∇β, V i E . (0.8) The left hand side of (0.8) is, by using (0.6) and the fact that J is an isome- try, P n

a,b=1 hD e

e a , ie b i

hE b , V i E . So we are led to the following computation, which uses first the fact that he a , ie b i

= 0, second the fact that i is a complex structure, and third the fact that the Lie bracket [e a , e b ] is tangent to Σ:

hD e

e a , ie b i

= − he a , iD e

e b i

= hie a , D e

e b i

= hie a , D e

e a + [e a , e b ]i

= hie a , D e

e a i

. So we have nh H, JV ~ i E = P n

a,b=1 hie a , D e

e a i

hE b , V i E . Now the right hand side of (0.8) can be computed as follows.. We derivate both sides of the relation e ^iβ = Θ

u ( u, e b 1 , · · · , e n ) with respect to e b and use the decomposition of D e

e a in the R -orthonormal basis ( u, e b 1 , · · · , e n , i u, ie b 1 , · · · , ie n ):

ie ^iβ D e

β = D e

e ^iβ

= D e

(Θ( b u, e 1 , · · · , e n ))

= Θ(e b , e 1 , · · · , e n ) + X n

a=1

Θ( u, e b 1 , · · · , D e

e a , · · · , e n )

= X n

a=1

Θ( u, e b 1 , · · · , e a , · · · , e n ) he a , D e

e a i

+ i hie a , D e

e a i

= ie ^iβ X n

a=1

hie a , D e

e a i

.

Here we used in the last line the fact that |e a | ²

= 1 = ⇒ he a , D e

e a i

= 0.

Hence we are left with D e

β = P n

a=1 hie a , D e

e a i

, which implies that h∇β, V i _E =

X n

b=1

D e

β hE b , V i _E = X n

a,b=1

hie a , D e

e a i

hE b , V i _E .

So we have proved (0.8).

A Hamiltonian stationary Lagrangian submanifold Σ in C P ⁿ is a Lagrangian sub- manifold which is a critical point of the n-volume functional A under first variations which are Hamiltonian vector fields with compact support. This means that for any smooth function with compact support h ∈ C _c

( C P ⁿ , R ), we have

δA ξ

(Σ) :=

Z

Σ

D H, ξ ~ h

E

E dvol = 0,

where ξ h is the Hamiltonian vector field of h, i.e. satisfies ω(ξ h , ·) + dh = 0 or ξ h = J ∇h. We also remark that if f ∈ C c

(Σ, R ), then there exist smooth extensions with compact support h of f , i.e. functions h ∈ C c

( C P ⁿ , R ) such that h

= f , and moreover the normal component of (ξ h )

does not depend on the choice of the extension h (it coincides actually with J ∇f , where ∇ is here the gradient with respect to the induced metric on Σ). So we deduce from Lemma 0.2 that actually δA ξ

(Σ) = _n ¹ R

Σ h∇β, ∇f i _E dvol. This implies the following.

Corollary 0.3. Any Lagrangian submanifold Σ in C P ⁿ is Hamiltonian sta- tionary if and only if β is a harmonic function on Σ, i.e.

∆ Σ β = 0.

The theory above extends to non simply connected surfaces Σ with the follow- ing restrictions. Let γ be a homotopically non trivial loop. The Legendrian lift of γ needs not close, so that in general its endpoints p 1 , p 2 ∈ S ²ⁿ⁺¹ are mul- tiples of each other by a factor e ^iθ . The same holds for the Lagrangian angle:

β(p 2 ) ≡ β(p 1 ) + (n + 1)θ mod 2π (since the tangent plane is also shifted by the Decktransformation z 7−→ e ^iθ z). In particular β is not always globally defined on surfaces in C P ⁿ with non trivial topology, unless the Legendrian lift is globally defined in S ²ⁿ⁺¹ / Z _n+1 (here Z _n+1 stands for the cubic roots of unity in SU (n + 1)).

References

[1] F. H´ elein , P. Romon , Hamiltonian stationary tori in the complex projective

plane, Proc. London Roy. Soc. (3) 90 (2005), 472–496.