Functionals on the space of almost complex stuctures
aa r X i v : . [ m a t h . DG ] J a n Functionals on the space of almostcomplex structures
Gabriella Clemente
Abstract
We study functionals on the space of almost complex structureson a compact C -manifold, whose variational properties could be usedto tackle Yau’s Challenge. Contents
Introduction 1Coordinates 4Euler-Lagrange equations 8
Introduction
This is supposed to be a step in the direction of understanding Yau’s Chal-lenge, which is to determine if there are compact almost complex man-ifolds of dimension at least 3 that cannot be given an integrable almostcomplex structure [3], through the calculus of variations. S-T. Yau pro-posed devising a parabolic flow on the space of almost complex structuresto study this question [2].Let X be a real 2 n -dimensional compact manifold, and AC ( X ) = { J ∈ C ∞ ( X, End C ( T X )) | J = − I d } be the space of almost complex structures on X. This is an almost complex Fr´echet manifold, and for any J ∈ AC ( X ) ,T Ac ( X ) ,J = { h ∈ C ∞ ( X, End C ( T X )) | J ◦ h + h ◦ J = 0 } , which can be seen fromthe identity 0 = dJ = dJ ◦ J + J ◦ dJ . An almost complex structure J : AC ( X ) → End( T AC ( X ) ) is given as J ( J )( u ) = J ◦ u, for any J ∈ AC ( X ) and1 ∈ T AC ( X ) ,J . Let g be a fixed Riemannian metric on X, and note that forany J ∈ AC ( X ) , we get an almost hermitian metric g J := (cid:16) g ( · , · ) + g ( J · , J · ) (cid:17) . We are looking for an energy functional F on AC ( X ) whose associatedgradient flow is a parabolic PDE. Ideally, the critical points of F shouldbe the integrable almost complex structures on X, and the Euler-Lagrangeequation of F should be elliptic so that the complex structures on X areenergy minimizers. We would then expect any solution of the flow equa-tion of F to converge to a genuine complex structure on X. In some specialcases, such as when AC ( X ) is connected (e.g. AC ( S )), the non-existenceof a flow solution might translate to the non-existence of complex struc-tures. A more thorough development of these ideas will be the subject offuture research. Here we only derive the Euler-Lagrange equations of thefunctionals N , e N : AC ( X ) → R ≥ , N ( J ) := Z X k N J k g J vol g , and e N ( J ) := Z X k N J k g J vol g J , where vol g is the Riemannian volume form, and vol g J is the volume formof ω := i ( g J − g J ) . Note that both N , and e N are identically zero on theintegrable structures. A very similar, real version of N appears in [1].We can think of first variations in terms of linear approximations. LetHerm (cid:16) Λ T , ∗ X ⊗ T , X (cid:17) denote the space of hermitian metrics on Λ T , ∗ X ⊗ T , X . Let f : AC ( X ) → C ∞ (cid:16) X, Λ T , ∗ X ⊗ T , X (cid:17) × Herm (cid:16) Λ T , ∗ X ⊗ T , X (cid:17) be the function f ( I ) = ( N , h ) = ( N ( I ) , h ( I )) = ( N I , g I − ∧ g I − ⊗ g I ) , and φ : C ∞ (cid:16) X, Λ T , ∗ X ⊗ T , X (cid:17) × Herm (cid:16) Λ T , ∗ X ⊗ T , X (cid:17) → C ∞ ( X, R ≥ )be the function φ ( N , h ) = h − ∧ h − ⊗ h ( N , N ) , and define ψ := φ ◦ f , where ψ ( I ) = g I − ∧ g I − ⊗ g I ( N I , N I ) =: k N I k g I . And now, let F : AC ( X ) → C ∞ (cid:16) X, Λ T , ∗ X ⊗ T , X (cid:17) × Herm (cid:16) Λ T , ∗ X ⊗ T , X (cid:17) × Ω n,n ( X ) ,F ( I ) = ( N ( I ) , h ( I ) , vol( I )) = ( N I , g I − ∧ g I − ⊗ g I , vol g I ) , and Φ : C ∞ (cid:16) X, Λ T , ∗ X ⊗ T , X (cid:17) × Herm (cid:16) Λ T , ∗ X ⊗ T , X (cid:17) × Ω n,n ( X ) → C ∞ ( X, R ≥ ) , ( N , h, vol) = h − ∧ h − ⊗ h ( N , N )vol h , and define Ψ := Φ ◦ F so that Ψ ( I ) = g I − ∧ g I − ⊗ g I ( N I , N I )vol g I =: k N I k g I vol g I . Let γ = (cid:16) g ( u · , J · ) + g ( J · , u · ) (cid:17) . Let J ∈ AC ( X ) , and δJ be a small perturbation of J ; i.e. if u is a nearby structurein AC ( X ) , then δJ = ( J + u ) − J .
Let δN = N ( J + δJ ) − N ( J ) = N J + u − N J = d J N J ( u ) + O ( u ) , δh = h ( J + δJ ) − h ( J ) = g J + u − g J = d J g J ( u ) + O ( u ) = γ + O ( u ) , and d vol = vol( J + δJ ) − vol( J ) = vol g J + u − vol g J = d J (vol g J )( u ) + O ( u ) = d J (vol g J )( u ) . Then, d J ( k N J k g J )( u ) ≈ [ φ ( N + δN , h + δh ) − φ ( N , h + δh )] + [ φ ( N , h + δh ) − φ ( N , h )] ≈ d N φ ( N , h + δh ) · δN + d h φ ( N , h ) · δh = d N J ( k N J k g J + u ) · d J N J ( u ) + d g J ( k N J k g J ) · γ , and d J ( k N J k g J vol g J )( u ) ≈ [ Φ ( N + δN , h + δh, vol + δ vol) − φ ( N , h + δh, vol + δ vol)]+[ Φ ( N , h + δh, vol + δ vol) − Φ ( N , h, vol + δ vol)]+[ Φ ( N , h, vol + δ vol) − Φ ( N , h, vol)] ≈ d N Φ ( N , h + δh, vol + δ vol) · δN + d h Φ ( N , h, vol + δ vol) · δh + d vol Φ ( N , h, vol) · δ vol= d N J ( k N J k g J + u vol g J + u ) · d J N J ( u ) + d g J ( k N J k g J vol g J + u ) · γ + d vol( k N J k g J vol g J ) · d J (vol g J )( u ) . The Nijenhuis tensor N J is g J -orthogonal to dN J ( u ) , and dN J ( u ) , . This is a consequence of the g J -orthogonality of the holomorphic, and an-tiholomorphic tangent bundles of X. From now on, all O ( u ) -terms will beomitted throughout with only a few exceptions in the last section. Then, wefind that d N ( k N J k g J + u ) · d J N J ( u ) = D d J N J ( u ) , N J E g J + u + D d J N J ( u ) , N J E g J + u = 2 ℜ h h dN , J ( u ) , N J i g J + u i . Proposition 1.
The first variation of N is d J N ( J )( u ) = Z X (cid:26) ℜ h h dN , J ( u ) , N J i g J + u i + d g J ( k N J k g J ) · γ (cid:27) vol g , nd that of e N is d J e N ( J )( u ) = Z X ℜ h h dN , J ( u ) , N J i g J + u i vol g J + u + Z X d g J ( k N J k g J + u vol g J + u ) · γ + Z X d vol ( k N J k g J vol g J ) · d J ( vol g J )( u ) . In order to retrieve the Euler-Lagrange equations of interest, we needto integrate 2 ℜ h h dN , J ( u ) , N J i g J + u i by parts. We do this in the coordinatesdefined below. Acknowledgment
I thank Jean-Pierre Demailly, my PhD supervisor,for his suggestions, and I thank the European Research Council for finan-cial support in the form of a PhD grant from the project “Algebraic andK¨ahler geometry” (ALKAGE, no. 670846).
Coordinates
We try to develop intrinsic complex coordinates on the almost hermitianmanifold (
X, J , g J ) , centered at a given point p ∈ X, that are the next bestalternative to both holomorphic coordinates, which exist only when J isintegrable, and geodesic coordinates at p, which exits i ff the fundamentalform ω of g J is K¨ahler.Recall that if ( z k ) ≤ k ≤ n are holomorphic coordinates on U ⊂ X, then¯ ∂ J z k = 0 on some neighborhood U x ⊂ X of every x ∈ U .
We do not havethat in the almost complex case. However, we can design complex coor-dinates, which will be denoted here by w k , for which ¯ ∂ J w k is as close aspossible to being zero on each U x . First note that we can always find com-plex coordinates z k ∈ C ∞ ( U p , C ) , centered at p, such that ¯ ∂ J z k ( p ) = 0 . Then, (cid:16) dz k ( p ) (cid:17) ≤ k ≤ n = (cid:16) ∂ J z k ( p ) (cid:17) ≤ k ≤ n is a basis of ( T , X,p ) ∗ , and so (cid:16) dz k ( p ) (cid:17) ≤ k ≤ n = (cid:16) ∂ J z k ( p ) (cid:17) ≤ k ≤ n is a basis of ( T , X,p ) ∗ . Hence, we get a local frame (cid:16) ∂ J z k (cid:17) ≤ k ≤ n of ( T , X,p ) ∗ , and so ¯ ∂ J z k = X ≤ l ≤ n f kl ( z ) ∂ J z l , where f kl ∈ C ∞ ( U p , C ) , and f kl ( p ) = 0 . Note that if z k were holomorphic, allof the coe ffi cient functions f kl would be identically zero. Here, for every4 ≤ l ≤ n, each of these functions has a Taylor expansion f kl ( z ) = X | α | + | β |≤ N c klαβ z α ¯ z β + O ( | z | N +1 ) . Given that we will only di ff erentiate once, we may instead work withthe truncation f kl ( z ) = X ≤ j ≤ n ( a jkl z j + a ′ jkl ¯ z j ) + O ( | z | ) . Again, if the z k were holomorphic, we would in particular have that a jkl = a ′ jkl = 0 , for all 1 ≤ j, k, l ≤ n. We wish to emulate this situation ( a jkl = a ′ jkl =0) in the almost complex case. Concretely, we are looking for new coordi-nates that anihilate as many of the coe ffi cients a jkl , and a ′ jkl as possible. Tothat end, let w k = z k + X ≤ r,s ≤ n ( α krs z r z s + β krs z r ¯ z s + γ krs ¯ z r ¯ z s ) + O ( | z | ) . We still have that w k ( p ) = 0 , ¯ ∂ J w k ( p ) = 0 , and dw k ( p ) = dz k ( p ) , and we stillget a local frame (cid:16) ∂ J w k (cid:17) ≤ k ≤ n of ( T , X | U p ) ∗ so that¯ ∂ J w k = X ≤ j,l ≤ n ( b jkl w j + b ′ jkl ¯ w j + O ( | w | )) ∂ J w l . We will see that the holomorphic condition prescribes β klm , and γ klm , whilethe geodesic condition can be used to solve for α klm . The point is that weare reducing the problem of finding optimal complex coordinates on analmost hermitian manifold to finding α krs , β krs , γ krs that anihilate the max-imum number of coe ffi cients of the Taylor expansions of f kl , and ω λ ¯ µ . Wecall w k = z k + P ≤ r,s ≤ n ( α krs z r z s + β krs z r ¯ z s + γ krs ¯ z r ¯ z s ) + O ( | z | ) , ≤ k ≤ n, with α krs , β krs , γ krs subject to these constraints almost holomorphic geodesic coor-dinates on X at p. Lemma 1.
Any complex coordinates z k ∈ C ∞ ( U p , C ) , ≤ k ≤ n, on an almosthermitian manifold ( X, J , g J ) that are centered at p, and for which ¯ ∂ J z k ( p ) = 0 , nd (cid:18) ∂∂z k ( p ) (cid:19) ≤ k ≤ n is an orthonormal basis of T , X,p , determine almost holomor-phic geodesic coordinates at p. Specifically, if the Taylor expansion of ¯ ∂ J z k on U p is ¯ ∂ J z k = X ≤ j,l ≤ n (cid:16) a kjl z j + a ′ kjl ¯ z j + O ( | z | ) (cid:17) ∂ J z l , and if ω m ¯ l = δ ml + n X s =1 ( τ m ¯ ls z s + τ ′ m ¯ l ¯ s ¯ z s ) + O ( | z | ) , then w k = z k − X ≤ m,l ≤ n h
14 ( a lkm + a mkl + τ l ¯ km + τ m ¯ kl ) z l z m + a klm z l ¯ z m + 14 ( a ′ klm + a ′ kml ) ¯ z l ¯ z m i + O ( | z | ) . Proof.
Since z m ¯ ∂ J z l = O ( | z | ) , ¯ z m ¯ ∂ J z l = O ( | z | ) , and ¯ ∂ J ¯ z l = ∂ J z l , and since γ klm is ( l, m )-symmetric,¯ ∂ J w k = ¯ ∂ J z k + X ≤ l,m ≤ n ¯ ∂ J (cid:16) α klm z l z m + β klm z l ¯ z m + γ klm ¯ z l ¯ z m ) + O ( | z | ) (cid:17) = ¯ ∂ J z k + X ≤ l,m ≤ n (cid:16) β klm z l ∂ J z m + ( γ klm + γ kml ) ¯ z l ∂ J z m (cid:17) + O ( | z | )= X ≤ l,m ≤ n (cid:16) a klm z l + a ′ klm ¯ z l (cid:17) ∂ J z m + X ≤ l,m ≤ n (cid:16) β klm z l ∂ J z m + 2 γ klm ¯ z l ∂ J z m (cid:17) + O ( | z | )= X ≤ l,m ≤ n h ( a klm + β klm ) z l + (cid:16) a ′ klm + 2 γ klm (cid:17) ¯ z l i ∂ J z m + O ( | z | ) . Based on this calculation, α klm is free to be any complex number, while β klm = − a klm . And we may take, at best, the symmetric part of a ′ klm + 2 γ klm to be zero, which is achieved by setting γ klm = − ( a ′ klm + a ′ kml ) . So far, wegathet that w k = z k + X ≤ m,l ≤ n h α klm z l z m − a klm z l ¯ z m −
14 ( a ′ klm + a ′ kml ) ¯ z l ¯ z m i + O ( | z | ) . Next, we optimize α klm subject to the constraint of w k being geodesic co-ordinates at p. Since α mlj is ( l, j )-symmetric,6 J w m = ∂ J z m + n X l,j =1 (2 α mlj z j + β mlj ¯ z j ) ∂ J z l + O ( | z | )so that ∂ J w m = ∂ J z m + n X l,j =1 (2 α mlj ¯ z j + β mlj z j ) ∂ J z l + O ( | z | ) . Then, since O ( | w | ) = O ( | z | ) ,i n X m,l =1 (cid:16) δ ml + O ( | w | ) (cid:17) ∂ J w m ∧ ∂ J w l = i n X m =1 ∂ J w m ∧ ∂ J w m + O ( | w | )= i n X m =1 (cid:20) ∂ J z m ∧ ∂ J z m + n X l,j =1 (2 α mlj ¯ z j + β mlj z j ) ∂ J z m ∧ ∂ J z l + n X l,j =1 (2 α lmj z j + β lmj ¯ z j ) ∂ J z m ∧ ∂ J z l (cid:21) + O ( | z | ) + O ( | w | )= i n X l,m =1 (cid:18) δ ml + n X j =1 h ( β mlj + 2 α lmj ) z j +2 α mlj + β lmj ) ¯ z j i + O ( | z | ) (cid:19) ∂ J z m ∧ ∂ J z l = i n X l,m =1 (cid:18) δ ml + n X j =1 ( τ m ¯ lj z j + τ ′ m ¯ l ¯ j ¯ z j ) + O ( | z | ) (cid:19) ∂ J z m ∧ ∂ J z l = ω, i.e. ω = i P nm,l =1 (cid:16) δ ml + O ( | w | ) (cid:17) ∂ J w m ∧ ∂ J w l , is a condition that can be at-tained, at best, by setting the ( m, j )-symmetric part of α lmj + ( β mlj − τ m ¯ lj )equal to zero. Thus, we may take α lmj = −
14 ( β mlj + β jlm − τ m ¯ lj − τ j ¯ lm )= 14 ( a mlj + a jlm + τ m ¯ lj + τ j ¯ lm ) , w k = z k + X ≤ m,l ≤ n h
14 ( a lkm + a mkl + τ l ¯ km + τ m ¯ kl ) z l z m − a klm z l ¯ z m −
14 ( a ′ klm + a ′ kml ) ¯ z l ¯ z m i + O ( | z | ) . Euler-Lagrange equations
Let ( w k ) nk =1 be almost holomorphic geodesic coordinates at p. We now havelocal coordinate frames (cid:18) ∂ , ∂w k (cid:19) ≤ k ≤ n of T , X , and (cid:18) ∂ , ∂ ¯ w k (cid:19) ≤ k ≤ n of T , X with dualcoframes (cid:16) dw , k (cid:17) ≤ k ≤ n , and (cid:16) d ¯ w , k (cid:17) ≤ k ≤ n . We also have the local coordinateexpressions( g J ) λ ¯ µ = (cid:28) ∂ , ∂w λ , ∂ , ∂ ¯ w µ (cid:29) g J = δ λµ + n X m =1 ( τ λ ¯ µm w m + τ ′ λ ¯ µm ¯ w m ) + O ( | w | ) ,N J = n X i,j,k =1 N k ¯ i ¯ j d ¯ w , i ∧ d ¯ w , j ⊗ ∂ , ∂w k , and likewise dN , J ( u ) = n X i,j,k =1 (cid:16) dN J ( u ) (cid:17) k ¯ i ¯ j d ¯ w , i ∧ d ¯ w , j ⊗ ∂ , ∂w k . Here we write dV = (cid:16) i (cid:17) n dw , ∧ d ¯ w , ∧ · · · ∧ dw , n ∧ d ¯ w , n , h := g J + u = g J + γ + O ( u ) , and h i ¯ j = δ ij + n X m =1 ( τ i ¯ jm w m + τ ′ i ¯ jm ¯ w m ) + O ( | w | ) + γ i ¯ j + O ( u ) , h − then being h i ¯ j := δ ij − n X m =1 ( τ i ¯ jm ¯ w m + τ ′ i ¯ jm w m ) − γ i ¯ j + n X c,v =1 ( τ i ¯ cv ¯ w v + τ ′ i ¯ cv w v ) γ c ¯ j + n X c,v =1 γ i ¯ c ( τ c ¯ jv ¯ w v + τ ′ c ¯ jv w v ) + O ( | w | ) + O ( u ) . Lemma 2. ∂ , h r ¯ s ∂w i ( p ) = τ r ¯ si + O ( u ) , ∂ , h r ¯ s ∂ ¯ w i ( p ) = τ ′ r ¯ si + O ( u ) ,∂ , h r ¯ s ∂w i ( p ) = − τ ′ r ¯ si + O ( u ) , and ∂ , h r ¯ s ∂ ¯ w i ( p ) = − τ r ¯ si + O ( u ) . Proof.
These equalities are a consequence of γ i ¯ j , and hence its derivativeswith respect to w m and ¯ w m , being of order O ( u ) . Lemma 3. (cid:28) dN , J ( u ) , N J (cid:29) g J + u = 2 (cid:18) h s ¯ m h j ¯ p h k ¯ q J i ¯ j ∂ , u k ¯ s ∂w i + h i ¯ m h j ¯ p h k ¯ q (cid:18) ∂ , u s ¯ j ∂ ¯ w i J ks + ∂ , u ¯ s ¯ j ∂ ¯ w i J k ¯ s (cid:19) + h s ¯ m h j ¯ p h k ¯ q ∂ , u k ¯ s ∂ ¯ w i J ¯ i ¯ j (cid:19) N q ¯ m ¯ p − h s ¯ m h j ¯ p h k ¯ q (cid:18) u i ¯ s ∂ , J k ¯ j ∂w i + u ¯ i ¯ s ∂ , J k ¯ j ∂ ¯ w i (cid:19) N q ¯ m ¯ p +2 h i ¯ m h j ¯ p h k ¯ q (cid:28) u (cid:20) ∂ , ∂ ¯ w i , J ∂ , ∂ ¯ w j (cid:21) , dw , k (cid:29) N q ¯ m ¯ p . Proof.
First, note that dN J ( u )( ζ, η ) = u [ ζ, J η ] + J [ ζ, uη ] + u [ J ζ, η ] + J [ uζ, η ] − [ uζ, J η ] − [ J ζ, uη ] . Since u = u sr dw , r ⊗ ∂ , ∂w s + u ¯ sr dw , r ⊗ ∂ , ∂ ¯ w s + u s ¯ r d ¯ w , r ⊗ ∂ , ∂w s + u ¯ s ¯ r d ¯ w , r ⊗ ∂ , ∂ ¯ w s , and J = J vt dw , t ⊗ ∂ , ∂w v + u ¯ vt dw , t ⊗ ∂ , ∂ ¯ w v + u v ¯ t d ¯ w , t ⊗ ∂ , ∂w v + u ¯ v ¯ t d ¯ w , t ⊗ ∂ , ∂ ¯ w v , itfollows that J (cid:20) ∂ , ∂ ¯ w i , u ∂ , ∂ ¯ w j (cid:21) = (cid:18) ∂ , u s ¯ j ∂ ¯ w i J vs + ∂ , u ¯ s ¯ j ∂ ¯ w i J v ¯ s (cid:19) ∂ , ∂w v + (cid:18) ∂ , u s ¯ j ∂ ¯ w i J ¯ vs + ∂ , u ¯ s ¯ j ∂ ¯ w i J ¯ v ¯ s (cid:19) ∂ , ∂ ¯ w v , u (cid:20) ∂ , ∂ ¯ w i , J ∂ , ∂ ¯ w j (cid:21) = (cid:18) ∂ , J s ¯ j ∂ ¯ w i u vs + ∂ , J ¯ s ¯ j ∂ ¯ w i u v ¯ s (cid:19) ∂ , ∂w v + (cid:18) ∂ , J s ¯ j ∂ ¯ w i u ¯ vs + ∂ , J ¯ s ¯ j ∂ ¯ w i u ¯ v ¯ s (cid:19) ∂ , ∂ ¯ w v . A similar computation shows that (cid:20)
J ∂ , ∂ ¯ w j , u ∂ , ∂ ¯ w i (cid:21) = (cid:18) J v ¯ j ∂ , u s ¯ i ∂w v + J ¯ v ¯ j ∂ , u s ¯ i ∂ ¯ w v − u v ¯ i ∂ , J s ¯ j ∂w v − u ¯ v ¯ i ∂ , J s ¯ j ∂ ¯ w v (cid:19) ∂ , ∂w s + (cid:18) J v ¯ j ∂ , u ¯ s ¯ i ∂w v + J v ¯ j ∂ , u ¯ s ¯ i ∂ ¯ w v − u v ¯ i ∂ , J ¯ s ¯ j ∂w v − u ¯ v ¯ i ∂ , J ¯ s ¯ j ∂ ¯ w v (cid:19) ∂ , ∂ ¯ w s . Therefore, (cid:28) J (cid:20) ∂ , ∂ ¯ w i , u ∂ , ∂ ¯ w j (cid:21) + (cid:20) J ∂ , ∂ ¯ w j , u ∂ , ∂ ¯ w i (cid:21) , dw , k (cid:29) = (cid:18) ∂ , u s ¯ j ∂ ¯ w i J ks + ∂ , u ¯ s ¯ j ∂ ¯ w i J k ¯ s (cid:19) + (cid:18) J s ¯ j ∂ , u k ¯ i ∂w s + J ¯ s ¯ j ∂ , u k ¯ i ∂ ¯ w s − u s ¯ i ∂ , J k ¯ j ∂w s − u ¯ s ¯ i ∂ , J k ¯ j ∂ ¯ w s (cid:19) , and so (cid:28) dN , J ( u ) , N J (cid:29) g J + u = h i ¯ m h j ¯ p h k ¯ q (cid:16) dN J ( u ) (cid:17) k ¯ i ¯ j N q ¯ m ¯ p = h i ¯ m h j ¯ p h k ¯ q (cid:28) u (cid:20) ∂ , ∂ ¯ w i , J ∂ , ∂ ¯ w j (cid:21) + J (cid:20) ∂ , ∂ ¯ w i , u ∂ , ∂ ¯ w j (cid:21) + u (cid:20) J ∂ , ∂ ¯ w i , ∂ , ∂ ¯ w j (cid:21) + J (cid:20) u ∂ , ∂ ¯ w i , ∂ , ∂ ¯ w j (cid:21) − (cid:20) u ∂ , ∂ ¯ w i , J ∂ , ∂ ¯ w j (cid:21) − (cid:20) J ∂ , ∂ ¯ w i , u ∂ , ∂ ¯ w j (cid:21) , dw , k (cid:29) N q ¯ m ¯ p = 2 h i ¯ m h j ¯ p h k ¯ q (cid:28) J (cid:20) ∂ , ∂ ¯ w i , u ∂ , ∂ ¯ w j (cid:21) + (cid:20) J ∂ , ∂ ¯ w j , u ∂ , ∂ ¯ w i (cid:21) , dw , k (cid:29) N q ¯ m ¯ p +2 h i ¯ m h ¯ jp h k ¯ q (cid:28) u (cid:20) ∂ , ∂ ¯ w i , J ∂ , ∂ ¯ w j (cid:21) , dw , k (cid:29) N q ¯ m ¯ p = 2 (cid:18) h s ¯ m h j ¯ p h k ¯ q J i ¯ j ∂ , u k ¯ s ∂w i + h i ¯ m h j ¯ p h k ¯ q (cid:18) ∂ , u s ¯ j ∂ ¯ w i J ks + ∂ , u ¯ s ¯ j ∂ ¯ w i J k ¯ s (cid:19) + h s ¯ m h j ¯ p h k ¯ q ∂ , u k ¯ s ∂ ¯ w i J ¯ i ¯ j (cid:19) N q ¯ m ¯ p − h s ¯ m h j ¯ p h k ¯ q (cid:18) u i ¯ s ∂ , J k ¯ j ∂w i + u ¯ i ¯ s ∂ , J k ¯ j ∂ ¯ w i (cid:19) N q ¯ m ¯ p +2 h i ¯ m h j ¯ p h k ¯ q (cid:28) u (cid:20) ∂ , ∂ ¯ w i , J ∂ , ∂ ¯ w j (cid:21) , dw , k (cid:29) N q ¯ m ¯ p . emma 4. At p, we have that ∂ , ∂w i (cid:20) h s ¯ m h j ¯ p h k ¯ q J i ¯ j N q ¯ m ¯ p det ( h ) (cid:21) u k ¯ s = (cid:18) − τ ′ s ¯ mi J i ¯ j N k ¯ m ¯ j − τ ′ j ¯ pi J i ¯ j N k ¯ s ¯ p + τ k ¯ qi J i ¯ j N q ¯ s ¯ j + ∂ , J i ¯ j ∂w i N k ¯ s ¯ j + J i ¯ j ∂ , N k ¯ s ¯ j ∂w i + J i ¯ j N k ¯ s ¯ j (cid:16) n X c =1 τ c ¯ ci (cid:17)(cid:19) u k ¯ s ,∂ , ∂ ¯ w i (cid:20) h i ¯ m h j ¯ p h k ¯ q J ks N q ¯ m ¯ p det ( h ) (cid:21) u s ¯ j = (cid:18) − τ i ¯ mi J ks N k ¯ m ¯ j − τ j ¯ pi J ks N k ¯ i ¯ p + τ ′ k ¯ qi J ks N q ¯ i ¯ j + ∂ , J ks ∂ ¯ w i N k ¯ i ¯ j + J ks ∂ , N k ¯ i ¯ j ∂ ¯ w i + J ks N k ¯ i ¯ j (cid:16) n X c =1 τ ′ c ¯ ci (cid:17)(cid:19) u s ¯ j ,∂ , ∂ ¯ w i (cid:20) h i ¯ m h j ¯ p h k ¯ q J k ¯ s N q ¯ m ¯ p det ( h ) (cid:21) u ¯ s ¯ j = (cid:18) − τ i ¯ mi J k ¯ s N k ¯ m ¯ j − τ j ¯ pi J k ¯ s N k ¯ i ¯ p + τ ′ k ¯ qi J k ¯ s N q ¯ i ¯ j + ∂ , J k ¯ s ∂ ¯ w i N k ¯ i ¯ j + J k ¯ s ∂ , N k ¯ i ¯ j ∂ ¯ w i + J k ¯ s N k ¯ i ¯ j (cid:16) n X c =1 τ ′ c ¯ ci (cid:17)(cid:19) u ¯ s ¯ j , and ∂ , ∂ ¯ w i (cid:20) h s ¯ m h j ¯ p h k ¯ q J ¯ i ¯ j N q ¯ m ¯ p det ( h ) (cid:21) u k ¯ s = (cid:18) − τ s ¯ mi J ¯ i ¯ j N k ¯ m ¯ j − τ j ¯ pi J ¯ i ¯ j N k ¯ s ¯ p + τ ′ k ¯ qi J ¯ i ¯ j N q ¯ s ¯ j + ∂ , J ¯ i ¯ j ∂ ¯ w i N k ¯ s ¯ j + J ¯ i ¯ j ∂ , N k ¯ s ¯ j ∂ ¯ w i + J ¯ i ¯ j N k ¯ s ¯ j (cid:16) n X c =1 τ ′ c ¯ ci (cid:17)(cid:19) u k ¯ s . Proof.
Note that h i ¯ j ( p ) = δ ij + γ i ¯ j ( p ) = δ ij + O ( u ) , and h i ¯ j ( p ) = δ ij − γ i ¯ j = δ ij + O ( u ) . Then using Lemma 2, ∂ , h s ¯ m ∂w i h j ¯ p h k ¯ q = − τ ′ s ¯ mi δ jp δ kq + O ( u ) , h s ¯ m ∂ , h j ¯ p ∂w i h k ¯ q = − τ ′ j ¯ pi δ sm δ kq + O ( u ) , h s ¯ m h j ¯ p ∂ , h k ¯ q ∂w i = τ k ¯ qi δ sm δ jp + O ( u ) . Now since det h ( p ) = 1 + P nc =1 γ c ¯ c = 1 + O ( u ) ,∂ , h s ¯ m h j ¯ p h k ¯ q ∂w i J i ¯ j N q ¯ m ¯ p det ( h ) u k ¯ s = (cid:18) ∂ , h s ¯ m ∂w i h j ¯ p h k ¯ q + h s ¯ m ∂ , h j ¯ p ∂w i h k ¯ q + h s ¯ m h j ¯ p ∂ , h k ¯ q ∂w i (cid:19) J i ¯ j N q ¯ m ¯ p det ( h ) u k ¯ s = (cid:16) − τ ′ s ¯ mi δ jp δ kq − τ ′ j ¯ pi δ sm δ kq + τ k ¯ qi δ sm δ jp (cid:17) J i ¯ j N q ¯ m ¯ p u k ¯ s . (1)Moreover, h s ¯ m h j ¯ p h k ¯ q ( p ) = δ sm δ jp δ kq + O ( u ) , and sincedet ( h ) = 1 + n X c,m =1 ( τ c ¯ cm w m + τ ′ c ¯ cm ¯ w m ) + n X c =1 γ c ¯ c + O ( | w | ) ,∂ , det ( h ) ∂w i ( p ) = n X c =1 τ c ¯ ci + n X c =1 ∂ , γ c ¯ c w i ( p ) = n X c =1 τ c ¯ ci + O ( u ) , implying that h s ¯ m h j ¯ p h k ¯ q ∂ , ∂w i (cid:18) J i ¯ j N q ¯ m ¯ p det ( h ) (cid:17) u k ¯ s = h s ¯ m h j ¯ p h k ¯ q (cid:18) ∂ , J i ¯ j ∂w i N q ¯ m ¯ p det ( h )+ J i ¯ j ∂ , N q ¯ m ¯ p ∂w i det ( h ) + J i ¯ j N q ¯ m ¯ p ∂ , det ( h ) ∂w i (cid:19) u k ¯ s = δ sm δ jp δ kq (cid:18) ∂ , J i ¯ j ∂w i N q ¯ m ¯ p + J i ¯ j ∂ , N q ¯ m ¯ p ∂w i + J i ¯ j N q ¯ m ¯ p (cid:16) n X c =1 τ c ¯ ci (cid:17)(cid:19) u k ¯ s . (2)Now, using equations 1 and 2, we see that12 , ∂w i (cid:20) h s ¯ m h j ¯ p h k ¯ q J i ¯ j N q ¯ m ¯ p det ( h ) (cid:21) u k ¯ s = (cid:20)(cid:16) − τ ′ s ¯ mi δ jp δ kq − τ ′ j ¯ pi δ sm δ kq + τ k ¯ qi δ sm δ jp (cid:17) J i ¯ j N q ¯ m ¯ p + δ sm δ jp δ kq (cid:18) ∂ , J i ¯ j ∂w i N q ¯ m ¯ p + J i ¯ j ∂ , N q ¯ m ¯ p ∂w i + J i ¯ j N q ¯ m ¯ p (cid:16) n X c =1 τ c ¯ ci (cid:17)(cid:19)(cid:21) u k ¯ s = (cid:18) − τ ′ s ¯ mi J i ¯ j N k ¯ m ¯ j − τ ′ j ¯ pi J i ¯ j N k ¯ s ¯ p + τ k ¯ qi J i ¯ j N q ¯ s ¯ j + ∂ , J i ¯ j ∂w i N k ¯ s ¯ j + J i ¯ j ∂ , N k ¯ s ¯ j ∂w i + J i ¯ j N k ¯ s ¯ j (cid:16) n X c =1 τ c ¯ ci (cid:17)(cid:19) u k ¯ s . Again, using Lemma 2, we find that ∂ , h i ¯ m h j ¯ p h k ¯ q ∂ ¯ w i J ks N q ¯ m ¯ p det ( h ) u s ¯ j = (cid:18) ∂ , h i ¯ m ∂ ¯ w i h j ¯ p h k ¯ q + h i ¯ m ∂ , h j ¯ p ∂ ¯ w i h k ¯ q + h i ¯ m h j ¯ p ∂ , h k ¯ q ∂ ¯ w i (cid:19) J ks N q ¯ m ¯ p det ( h ) u s ¯ j = (cid:16) − τ i ¯ mi δ jp δ kq − τ j ¯ pi δ im δ kq + τ ′ k ¯ qi δ im δ jp (cid:17) J ks N q ¯ m ¯ p u s ¯ j . Since ∂ , det ( h ) ∂ ¯ w i ( p ) = n X c =1 τ ′ c ¯ ci + O ( u ) ,h i ¯ m h j ¯ p h k ¯ q ∂ , ∂ ¯ w i (cid:18) J ks N q ¯ m ¯ p det ( h ) (cid:19) u s ¯ j = h i ¯ m h j ¯ p h k ¯ q (cid:18) ∂ , J ks ∂ ¯ w i N q ¯ m ¯ p det ( h )+ J ks ∂ , N q ¯ m ¯ p ∂ ¯ w i det ( h ) + J ks N q ¯ m ¯ p ∂ , det ( h ) ∂ ¯ w i (cid:19) u s ¯ j = δ im δ jp δ kq (cid:18) ∂ , J ks ∂ ¯ w i N q ¯ m ¯ p + J ks ∂ , N q ¯ m ¯ p ∂ ¯ w i + J ks N q ¯ m ¯ p (cid:16) n X c =1 τ ′ c ¯ ci (cid:17)(cid:19) u s ¯ j , (3)confirming that 13 , ∂ ¯ w i (cid:20) h i ¯ m h j ¯ p h k ¯ q J ks N q ¯ m ¯ p det ( h ) (cid:21) u s ¯ j = (cid:20)(cid:16) − τ i ¯ mi δ jp δ kq − τ j ¯ pi δ im δ kq + τ ′ k ¯ qi δ im δ jp (cid:17) J ks N q ¯ m ¯ p + δ im δ jp δ kq (cid:18) ∂ , J ks ∂ ¯ w i N q ¯ m ¯ p + J ks ∂ , N q ¯ m ¯ p ∂ ¯ w i + J ks N q ¯ m ¯ p (cid:16) n X c =1 τ ′ c ¯ ci (cid:17)(cid:19)(cid:21) u s ¯ j = (cid:18) − τ i ¯ mi J ks N k ¯ m ¯ j − τ j ¯ pi J ks N k ¯ i ¯ p + τ ′ k ¯ qi J ks N q ¯ i ¯ j + ∂ , J ks ∂ ¯ w i N k ¯ i ¯ j + J ks ∂ , N k ¯ i ¯ j ∂ ¯ w i + J ks N k ¯ i ¯ j (cid:16) n X c =1 τ ′ c ¯ ci (cid:17)(cid:19) u s ¯ j . Lemma 5.
Let g ′ ij = g (cid:16) ∂ , ∂w i , J ∂ , ∂w j (cid:17) , g ′ ¯ ij = g (cid:16) ∂ , ∂ ¯ w i , J ∂ , ∂w j (cid:17) , and so forth. Then, d g J (cid:16) k N k g J + u vol g J + u (cid:17) · γ (cid:16) p ) = − (cid:18) u ks (cid:16) g ′ k ¯ j N v ¯ i ¯ j N v ¯ i ¯ s − g ′ k ¯ m N s ¯ i ¯ j N m ¯ i ¯ j (cid:17) + u ¯ ks (cid:16) g ′ ¯ k ¯ j N v ¯ i ¯ j N v ¯ i ¯ s − g ′ ¯ k ¯ m N s ¯ i ¯ j N m ¯ i ¯ j (cid:17) + u k ¯ s (cid:16) g ′ km N v ¯ i ¯ s N v ¯ i ¯ m − g ′ kv N v ¯ i ¯ j N s ¯ i ¯ j (cid:17) + u ¯ k ¯ s (cid:16) g ′ ¯ km N v ¯ i ¯ s N v ¯ i ¯ m − g ′ ¯ kv N v ¯ i ¯ j N s ¯ i ¯ j (cid:17)(cid:19) dV , (4) and d vol (cid:16) k N J k vol g J (cid:17) · d J ( vol g J )( u )( p ) = (cid:18)(cid:16) u ks g ′ k ¯ s + u ¯ ks g ′ ¯ k ¯ s + u k ¯ s g ′ ks + u ¯ k ¯ s g ′ ¯ ks (cid:17) | N v ¯ i ¯ j | (cid:19) dV (5) Proof.
First note that d g J (cid:16) k N k g J + u vol g J + u (cid:17) · γ (cid:16) p ) = (cid:16) k N J k I + γ − k N J k I (cid:17)(cid:16) tr ( γ ) (cid:17) dV = − (cid:16) γ j ¯ m N k ¯ i ¯ j N k ¯ i ¯ m + γ i ¯ m N k ¯ i ¯ j N k ¯ m ¯ j − γ k ¯ m N k ¯ i ¯ j N m ¯ i ¯ j (cid:17) dV = − (cid:16) γ j ¯ m N k ¯ i ¯ j N k ¯ i ¯ m − γ k ¯ m N k ¯ i ¯ j N m ¯ i ¯ j (cid:17) dV , and that d vol (cid:16) k N J k vol g J (cid:17) · d J (vol g J )( u ) (cid:16) p ) = k N J k I (cid:16) vol I + γ − vol I (cid:17) = | N k ¯ i ¯ j | tr ( γ ) dV . γ k ¯ m = u vk g ′ v ¯ m + u ¯ vk g ′ ¯ v ¯ m + u v ¯ m g ′ vk + u ¯ v ¯ m g ′ ¯ vk . Equation 4 is obtained by arelabeling of indices in γ m ¯ j , γ m ¯ i , and γ k ¯ m so that the upper index of u is k (or ¯ k ) and the lower index is s (or ¯ s ), and collecting terms with thesame u -coe ffi cients. Equation 5 follows after writing tr ( γ ) = P ns =1 γ s ¯ s = u vs g ′ v ¯ s + u ¯ vs g ′ ¯ v ¯ s + u v ¯ s g ′ vs + u ¯ v ¯ s g ′ ¯ vs , and a similar relabelling of indices. Proposition 2.
Suppose that u is compactly supported in U p . Let ≤ p, q ≤ n. The Euler-Lagrange system of equations of e N at p is e T qp := 4 ∂ , J p ¯ j ∂ ¯ w i N q ¯ i ¯ j − (cid:16) g ′ q ¯ j N v ¯ i ¯ j N v ¯ i ¯ p − g ′ q ¯ m N p ¯ i ¯ j N m ¯ i ¯ j (cid:17) + g ′ q ¯ p | N v ¯ i ¯ j | = 0 , e T ¯ qp := − (cid:16) g ′ ¯ q ¯ j N v ¯ i ¯ j N v ¯ i ¯ p − g ′ ¯ q ¯ m N p ¯ i ¯ j N m ¯ i ¯ j (cid:17) + g ′ ¯ q ¯ p | N v ¯ i ¯ j | = 0 , e T q ¯ p := 4 (cid:20)(cid:18) J (cid:18) ∂ , ∂ ¯ w j (cid:19) ω m ¯ p (cid:19) N q ¯ m ¯ j + (cid:18) J (cid:18) ∂ , ∂ ¯ w j (cid:19) ω m ¯ j (cid:19) N q ¯ p ¯ m − (cid:18) J (cid:18) ∂ , ∂ ¯ w j (cid:19) ω q ¯ m (cid:19) N m ¯ p ¯ j − ∂ , ∂w i (cid:16) J i ¯ j N q ¯ p ¯ j (cid:17) − ∂ , ∂ ¯ w i (cid:16) J ¯ i ¯ j N q ¯ p ¯ j (cid:17) − (cid:18) J (cid:18) ∂ , ∂ ¯ w j (cid:19)(cid:18) n X c =1 ω c ¯ c (cid:19)(cid:19) N q ¯ p ¯ j + J jq (cid:18) ∂ , ω m ¯ i ∂ ¯ w i N j ¯ m ¯ p + ∂ , ω m ¯ p ∂ ¯ w i N j ¯ i ¯ m − ∂ , ω j ¯ m ∂ ¯ w i N m ¯ i ¯ p − n X c =1 ∂ , ω c ¯ c ∂ ¯ w i N j ¯ i ¯ p (cid:19) − ∂ , ∂ ¯ w i (cid:16) J jq N j ¯ i ¯ p (cid:17) − ∂ , J i ¯ j ∂w q N i ¯ p ¯ j + ∂ , J ¯ p ¯ j ∂ ¯ w i N q ¯ i ¯ j (cid:21) − (cid:16) g ′ qm N v ¯ i ¯ p N v ¯ i ¯ m − g ′ qv N v ¯ i ¯ j N p ¯ i ¯ j (cid:17) + g ′ qp | N v ¯ i ¯ j | = 0 , e T ¯ q ¯ p := 4 (cid:20) J j ¯ q (cid:18) ∂ , ω m ¯ i ∂ ¯ w i N j ¯ m ¯ p + ∂ , ω m ¯ p ∂ ¯ w i N j ¯ i ¯ m − ∂ , ω j ¯ m ∂ ¯ w i N m ¯ i ¯ p − n X c =1 ∂ , ω c ¯ c ∂ ¯ w i N m ¯ p ¯ j (cid:19) − ∂ , ∂ ¯ w i (cid:16) J j ¯ q N j ¯ i ¯ p (cid:17) − ∂ , J ¯ i ¯ j ∂ ¯ w q N i ¯ p ¯ j (cid:21) − (cid:16) g ′ ¯ qm N v ¯ i ¯ p N v ¯ i ¯ m − g ′ ¯ qv N v ¯ i ¯ j N p ¯ i ¯ j (cid:17) + g ′ ¯ qp | N v ¯ i ¯ j | = 0 , and that of N is e T qp − g ′ q ¯ p | N v ¯ i ¯ j | = 0 , e T ¯ qp − g ′ ¯ q ¯ p | N v ¯ i ¯ j | = 0 , T q ¯ p + 4 (cid:20)(cid:18) J (cid:18) ∂ , ∂ ¯ w j (cid:19)(cid:18) n X c =1 ω c ¯ c (cid:19)(cid:19) N q ¯ p ¯ j + J jq n X c =1 ∂ , ω c ¯ c ∂ ¯ w i N j ¯ i ¯ p (cid:21) − g ′ qp | N v ¯ i ¯ j | = 0 , e T ¯ q ¯ p + 4 J j ¯ q n X c =1 ∂ , ω c ¯ c ∂ ¯ w i N m ¯ p ¯ j − g ′ ¯ qp | N v ¯ i ¯ j | = 0 . Proof.
The procedure is to use Lemma 4 to integrate by parts the termsinvolving derivatives of u in the first variation of e N (Proposition 1), andthen isolate u in the resulting formula by writing it as the g J -inner productof a tensor, the Euler-Lagrange equation at p, and u. Lemmas 3, 5 lead to d J e N ( J )( u ) = 4 ℜ (cid:20) Z X (cid:18) h s ¯ m h j ¯ p h k ¯ q J i ¯ j ∂ , u k ¯ s ∂w i + h i ¯ m h j ¯ p h k ¯ q (cid:18) ∂ , u s ¯ j ∂ ¯ w i J ks + ∂ , u ¯ s ¯ j ∂ ¯ w i J k ¯ s (cid:19) + h s ¯ m h j ¯ p h k ¯ q ∂ , u k ¯ s ∂ ¯ w i J ¯ i ¯ j (cid:19) N q ¯ m ¯ p vol h (cid:21) ( p ) − ℜ (cid:20) Z X h s ¯ m h j ¯ p h k ¯ q (cid:18) u i ¯ s ∂ , J k ¯ j ∂w i + u ¯ i ¯ s ∂ , J k ¯ j ∂ ¯ w i (cid:19) N q ¯ m ¯ p vol g J + u (cid:21) ( p )+4 ℜ (cid:20) Z X h i ¯ m h j ¯ p h k ¯ q (cid:28) u (cid:20) ∂ , ∂ ¯ w i , J ∂ , ∂ ¯ w j (cid:21) , dw , k (cid:29) N q ¯ m ¯ p vol g J + u (cid:21) ( p ) − Z X (cid:18) u ks (cid:16) g ′ k ¯ j N v ¯ i ¯ j N v ¯ i ¯ s − g ′ k ¯ m N s ¯ i ¯ j N m ¯ i ¯ j (cid:17) + u ¯ ks (cid:16) g ′ ¯ k ¯ j N v ¯ i ¯ j N v ¯ i ¯ s − g ′ ¯ k ¯ m N s ¯ i ¯ j N m ¯ i ¯ j (cid:17) + u k ¯ s (cid:16) g ′ km N v ¯ i ¯ s N v ¯ i ¯ m − g ′ kv N v ¯ i ¯ j N s ¯ i ¯ j (cid:17) + u ¯ k ¯ s (cid:16) g ′ ¯ km N v ¯ i ¯ s N v ¯ i ¯ m − g ′ ¯ kv N v ¯ i ¯ j N s ¯ i ¯ j (cid:17)(cid:19) dV + Z X (cid:16) u ks g ′ k ¯ s + u ¯ ks g ′ ¯ k ¯ s + u k ¯ s g ′ ks + u ¯ k ¯ s g ′ ¯ ks (cid:17) | N v ¯ i ¯ j | dV = 4 ℜ (cid:20) Z X (cid:18) τ ′ s ¯ mi J i ¯ j N k ¯ m ¯ j + τ ′ j ¯ pi J i ¯ j N k ¯ s ¯ p − τ k ¯ qi J i ¯ j N q ¯ s ¯ j − ∂ , J i ¯ j ∂w i N k ¯ s ¯ j − J i ¯ j ∂ , N k ¯ s ¯ j ∂w i − J i ¯ j N k ¯ s ¯ j (cid:16) n X c =1 τ c ¯ ci (cid:17)(cid:19) u k ¯ s dV + Z X (cid:18) τ i ¯ mi J ks N k ¯ m ¯ j + τ j ¯ pi J ks N k ¯ i ¯ p − τ ′ k ¯ qi J ks N q ¯ i ¯ j − ∂ , J ks ∂ ¯ w i N k ¯ i ¯ j − J ks ∂ , N k ¯ i ¯ j ∂ ¯ w i − J ks N k ¯ i ¯ j (cid:16) n X c =1 τ ′ c ¯ ci (cid:17)(cid:19) u s ¯ j dV + Z X (cid:18) τ i ¯ mi J k ¯ s N k ¯ m ¯ j +16 j ¯ pi J k ¯ s N k ¯ i ¯ p − τ ′ k ¯ qi J k ¯ s N q ¯ i ¯ j − ∂ , J k ¯ s ∂ ¯ w i N k ¯ i ¯ j − J k ¯ s ∂ , N k ¯ i ¯ j ∂ ¯ w i − J k ¯ s N k ¯ i ¯ j (cid:16) n X c =1 τ ′ c ¯ ci (cid:17)(cid:19) u ¯ s ¯ j dV + Z X (cid:18) τ s ¯ mi J ¯ i ¯ j N k ¯ m ¯ j + τ j ¯ pi J ¯ i ¯ j N k ¯ s ¯ p − τ ′ k ¯ qi J ¯ i ¯ j N q ¯ s ¯ j − ∂ , J ¯ i ¯ j ∂ ¯ w i N k ¯ s ¯ j − J ¯ i ¯ j ∂ , N k ¯ s ¯ j ∂ ¯ w i − J ¯ i ¯ j N k ¯ s ¯ j (cid:16) n X c =1 τ ′ c ¯ ci (cid:17)(cid:19) u k ¯ s dV (cid:21) − ℜ (cid:20) Z X (cid:18) u k ¯ s ∂ , J i ¯ j ∂w k + u ¯ k ¯ s ∂ , J i ¯ j ∂ ¯ w k (cid:19) N i ¯ s ¯ j dV (cid:21) + 4 ℜ (cid:20) Z X (cid:18) ∂ , J s ¯ j ∂ ¯ w i u ks + ∂ , J ¯ s ¯ j ∂ ¯ w i u k ¯ s (cid:19) N k ¯ i ¯ j dV (cid:21) − Z X (cid:18) u ks (cid:16) g ′ k ¯ j N v ¯ i ¯ j N v ¯ i ¯ s − g ′ k ¯ m N s ¯ i ¯ j N m ¯ i ¯ j (cid:17) + u ¯ ks (cid:16) g ′ ¯ k ¯ j N v ¯ i ¯ j N v ¯ i ¯ s − g ′ ¯ k ¯ m N s ¯ i ¯ j N m ¯ i ¯ j (cid:17) + u k ¯ s (cid:16) g ′ km N v ¯ i ¯ s N v ¯ i ¯ m − g ′ kv N v ¯ i ¯ j N s ¯ i ¯ j (cid:17) + u ¯ k ¯ s (cid:16) g ′ ¯ km N v ¯ i ¯ s N v ¯ i ¯ m − g ′ ¯ kv N v ¯ i ¯ j N s ¯ i ¯ j (cid:17)(cid:19) dV + Z X (cid:16) u ks g ′ k ¯ s + u ¯ ks g ′ ¯ k ¯ s + u k ¯ s g ′ ks + u ¯ k ¯ s g ′ ¯ ks (cid:17) | N v ¯ i ¯ j | dV = ℜ (cid:26) Z X (cid:20)(cid:18) (cid:18)(cid:16) τ ′ s ¯ mi J i ¯ j + τ s ¯ mi J ¯ i ¯ j (cid:17) N k ¯ m ¯ j + (cid:16) τ ′ j ¯ mi J i ¯ j + τ j ¯ mi J ¯ i ¯ j (cid:17) N k ¯ s ¯ m − (cid:16) τ k ¯ mi J i ¯ j + τ ′ k ¯ mi J ¯ i ¯ j (cid:17) N m ¯ s ¯ j − (cid:18) ∂ , J i ¯ j ∂w i + ∂ , J ¯ i ¯ j ∂ ¯ w i (cid:19) N k ¯ s ¯ j − J i ¯ j ∂ , N k ¯ s ¯ j ∂w i − J ¯ i ¯ j ∂ , N k ¯ s ¯ j ∂ ¯ w i − N k ¯ s ¯ j (cid:18) J i ¯ j (cid:16) n X c =1 τ c ¯ ci (cid:17) + J ¯ i ¯ j (cid:16) n X c =1 τ ′ c ¯ ci (cid:17)(cid:19) + τ i ¯ mi J jk N j ¯ m ¯ s + τ s ¯ mi J jk N j ¯ i ¯ m − τ j ¯ mi J jk N m ¯ i ¯ s − ∂ , J jk ∂ ¯ w i N j ¯ i ¯ s − J jk ∂ , N j ¯ i ¯ s ∂ ¯ w i − J jk N j ¯ i ¯ s (cid:16) n X c =1 τ ′ c ¯ ci (cid:17) − ∂ , J i ¯ j ∂w k N i ¯ s ¯ j + ∂ , J ¯ s ¯ j ∂ ¯ w i N k ¯ i ¯ j (cid:19) − (cid:16) g ′ km N v ¯ i ¯ s N v ¯ i ¯ m − g ′ kv N v ¯ i ¯ j N s ¯ i ¯ j (cid:17) + g ′ ks | N v ¯ i ¯ j | (cid:19) u k ¯ s + (cid:18) (cid:18) τ i ¯ mi J j ¯ k N j ¯ m ¯ s + τ s ¯ mi J j ¯ k N j ¯ i ¯ m − τ ′ j ¯ mi J j ¯ k N m ¯ i ¯ s − ∂ , J j ¯ k ∂ ¯ w i N j ¯ i ¯ s − J j ¯ k ∂ , N j ¯ i ¯ s ∂ ¯ w i − J j ¯ k N j ¯ i ¯ s (cid:16) n X c =1 τ ′ c ¯ ci (cid:17) − ∂ , J i ¯ j ∂ ¯ w k N i ¯ s ¯ j (cid:19) − (cid:16) g ′ ¯ km N v ¯ i ¯ s N v ¯ i ¯ m − g ′ ¯ kv N v ¯ i ¯ j N s ¯ i ¯ j (cid:17) +17 ′ ¯ ks | N v ¯ i ¯ j | (cid:19) u ¯ k ¯ s + (cid:18) − (cid:16) g ′ ¯ k ¯ j N v ¯ i ¯ j N v ¯ i ¯ s − g ′ ¯ k ¯ m N s ¯ i ¯ j N m ¯ i ¯ j (cid:17) + g ′ ¯ k ¯ s | N v ¯ i ¯ j | (cid:19) u ¯ ks + (cid:18) ∂ , J s ¯ j ∂ ¯ w i N k ¯ i ¯ j − (cid:16) g ′ k ¯ j N v ¯ i ¯ j N v ¯ i ¯ s − g ′ k ¯ m N s ¯ i ¯ j N m ¯ i ¯ j (cid:17) + g ′ k ¯ s | N v ¯ i ¯ j | (cid:19) u ks (cid:21) dV (cid:27) We may use g J ( p ) = I to induce the inner product h T , V i = P ns,k =1 ( T ks V ks + T ¯ ks V ¯ ks + T k ¯ s V k ¯ s + T ¯ k ¯ s V ¯ k ¯ s ) of any T , V ∈ C ∞ (cid:16) X, End( T C X,p ) (cid:17) . Consider now thetensor e T J = e T qp dw , p ⊗ ∂ , ∂w q + e T ¯ qp dw , p ⊗ ∂ , ∂ ¯ w q + e T q ¯ p d ¯ w , p ⊗ ∂ , ∂w q + e T ¯ q ¯ p d ¯ w , p ⊗ ∂ , ∂ ¯ w q , where e T qp = 4 ∂ , J p ¯ j ∂ ¯ w i N q ¯ i ¯ j − (cid:16) g ′ q ¯ j N v ¯ i ¯ j N v ¯ i ¯ p − g ′ q ¯ m N p ¯ i ¯ j N m ¯ i ¯ j (cid:17) + g ′ q ¯ p | N v ¯ i ¯ j | , e T ¯ qp = − (cid:16) g ′ ¯ q ¯ j N v ¯ i ¯ j N v ¯ i ¯ p − g ′ ¯ q ¯ m N p ¯ i ¯ j N m ¯ i ¯ j (cid:17) + g ′ ¯ q ¯ p | N v ¯ i ¯ j | , e T q ¯ p = 4 (cid:18)(cid:16) τ ′ p ¯ mi J i ¯ j + τ p ¯ mi J ¯ i ¯ j (cid:17) N q ¯ m ¯ j + (cid:16) τ ′ j ¯ mi J i ¯ j + τ j ¯ mi J ¯ i ¯ j (cid:17) N q ¯ p ¯ m − (cid:16) τ q ¯ mi J i ¯ j + τ ′ q ¯ mi J ¯ i ¯ j (cid:17) N m ¯ p ¯ j − (cid:18) ∂ , J i ¯ j ∂w i + ∂ , J ¯ i ¯ j ∂ ¯ w i (cid:19) N q ¯ p ¯ j − J i ¯ j ∂ , N q ¯ p ¯ j ∂w i − J ¯ i ¯ j ∂ , N q ¯ p ¯ j ∂ ¯ w i − N q ¯ p ¯ j (cid:18) J i ¯ j (cid:16) n X c =1 τ c ¯ ci (cid:17) + J ¯ i ¯ j (cid:16) n X c =1 τ ′ c ¯ ci (cid:17)(cid:19) + τ i ¯ mi J jq N j ¯ m ¯ p + τ p ¯ mi J jq N j ¯ i ¯ m − τ ′ j ¯ mi J jq N m ¯ i ¯ p − ∂ , J jq ∂ ¯ w i N j ¯ i ¯ p − J jq ∂ , N j ¯ i ¯ p ∂ ¯ w i − J jq N j ¯ i ¯ p (cid:16) n X c =1 τ ′ c ¯ ci (cid:17) − ∂ , J i ¯ j ∂w q N i ¯ p ¯ j + ∂ , J ¯ p ¯ j ∂ ¯ w i N q ¯ i ¯ j (cid:19) − (cid:16) g ′ qm N v ¯ i ¯ p N v ¯ i ¯ m − g ′ qv N v ¯ i ¯ j N p ¯ i ¯ j (cid:17) + g ′ qp | N v ¯ i ¯ j | , e T ¯ q ¯ p = 4 (cid:18) τ i ¯ mi J j ¯ q N j ¯ m ¯ p + τ p ¯ mi J j ¯ q N j ¯ i ¯ m − τ ′ j ¯ mi J j ¯ q N m ¯ i ¯ p − ∂ , J j ¯ q ∂ ¯ w i N j ¯ i ¯ p − J j ¯ q ∂ , N j ¯ i ¯ p ∂ ¯ w i − J j ¯ q N j ¯ i ¯ p (cid:16) n X c =1 τ ′ c ¯ ci (cid:17) − ∂ , J i ¯ j ∂ ¯ w q N i ¯ p ¯ j (cid:19) − (cid:16) g ′ ¯ qm N v ¯ i ¯ p N v ¯ i ¯ m − g ′ ¯ qv N v ¯ i ¯ j N p ¯ i ¯ j (cid:17) + g ′ ¯ qp | N v ¯ i ¯ j | . Then, d J e N ( J )( u ) = ℜ (cid:26) Z X h e T J , u i I dV (cid:27) , and so e T J = 0 is the Euler-Lagrange equation at p. We can rewrite T ¯ qp , and T q ¯ p somewhat more meaningfully, using that J (cid:18) ∂ , ∂ ¯ w j (cid:19) = J i ¯ j ∂ , ∂w i + J ¯ i ¯ j ∂ , ∂ ¯ w i , and the hermitian nature of the fundamental form ω : T ¯ q ¯ p = 4 (cid:20) J j ¯ q (cid:18) ∂ , ω m ¯ i ∂ ¯ w i N j ¯ m ¯ p + ∂ , ω m ¯ p ∂ ¯ w i N j ¯ i ¯ m − ∂ , ω j ¯ m ∂ ¯ w i N m ¯ i ¯ p − n X c =1 ∂ , ω c ¯ c ∂ ¯ w i N m ¯ p ¯ j (cid:19) − ∂ , ∂ ¯ w i (cid:16) J j ¯ q N j ¯ i ¯ p (cid:17) − ∂ , J ¯ i ¯ j ∂ ¯ w q N i ¯ p ¯ j (cid:21) − (cid:16) g ′ ¯ qm N v ¯ i ¯ p N v ¯ i ¯ m − g ′ ¯ qv N v ¯ i ¯ j N p ¯ i ¯ j (cid:17) + g ′ ¯ qp | N v ¯ i ¯ j | , T q ¯ p = 4 (cid:20)(cid:18) J (cid:18) ∂ , ∂ ¯ w j (cid:19) ω m ¯ p (cid:19) N q ¯ m ¯ j + (cid:18) J (cid:18) ∂ , ∂ ¯ w j (cid:19) ω m ¯ j (cid:19) N q ¯ p ¯ m − (cid:18) J (cid:18) ∂ , ∂ ¯ w j (cid:19) ω q ¯ m (cid:19) N m ¯ p ¯ j − ∂ , ∂w i (cid:16) J i ¯ j N q ¯ p ¯ j (cid:17) − ∂ , ∂ ¯ w i (cid:16) J ¯ i ¯ j N q ¯ p ¯ j (cid:17) − (cid:18) J (cid:18) ∂ , ∂ ¯ w j (cid:19)(cid:18) n X c =1 ω c ¯ c (cid:19)(cid:19) N q ¯ p ¯ j + J jq (cid:18) ∂ , ω m ¯ i ∂ ¯ w i N j ¯ m ¯ p + ∂ , ω m ¯ p ∂ ¯ w i N j ¯ i ¯ m − ∂ , ω j ¯ m ∂ ¯ w i N m ¯ i ¯ p − n X c =1 ∂ , ω c ¯ c ∂ ¯ w i N j ¯ i ¯ p (cid:19) − ∂ , ∂ ¯ w i (cid:16) J jq N j ¯ i ¯ p (cid:17) − ∂ , J i ¯ j ∂w q N i ¯ p ¯ j + ∂ , J ¯ p ¯ j ∂ ¯ w i N q ¯ i ¯ j (cid:21) − (cid:16) g ′ qm N v ¯ i ¯ p N v ¯ i ¯ m − g ′ qv N v ¯ i ¯ j N p ¯ i ¯ j (cid:17) + g ′ qp | N v ¯ i ¯ j | . In the case of N , there is no d vol( k N J k g J vol g J ) · d J (vol g J )( u ) term in the firstvariation, neither are there derivatives of the Riemannian volume form.We can recover the Euler-Lagrange equation of N at p from that of e N tofind that it is a tensor equation T N = 0 , where the components of T N areof the claimed form. 19ll integrable almost complex structures on X are critical points ofboth N , and e N , but they are likely not the only ones. It is also unclear if e N has any advantages over N . It might make sense to try eliminating thederivatives of N J that appear in the Euler-Lagrange equation of N sincethey do not directly contain information about integrability. References [1] J. Milewski,
Holomorphons and the standard almost complex structureon S . Annales Societatis Mathematicae Polonae, Series I: Commenta-tiones Mathematicae, XLVI (2) (2006), 245–254.[2] S-T. Yau,