aa r X i v : . [ m a t h . DG ] S e p TWISTORIAL CONSTRUCTION OF MINIMALHYPERSURFACES
JOHANN DAVIDOV
Abstract.
Every almost Hermitian structure ( g, J ) on a four-manifold M de-termines a hypersurface Σ J in the (positive) twistor space of ( M, g ) consistingof the complex structures anti-commuting with J . In this note we find theconditions under which Σ J is minimal with respect to a natural Riemannianmetric on the twistor space in the cases when J is integrable or symplectic.Several examples illustrating the obtained results are also discussed. Keywords:
Twistor spaces; minimal hypersurfaces.
Mathematics Subject Classification 2010 . Primary: 53C28; Secondary: 53A10,49Q05. Introduction
The twistor space Z of a Riemannian manifold ( M, g ) is the bundle on M parametrizing the complex structures on the tangent spaces of M compatible withthe metric g . Thus the almost Hermitian structures on ( M, g ) are sections of Z .Given such a structure J , we can consider the hypersurface Σ J of points of Z rep-resenting complex structures anti-commuting with J . The twistor space admitsa 1-parameter family h t of Riemannian metrics, the so-called canonical variationof g . Then it is natural to relate geometric properties of the hypersurface Σ J inthe Riemannian manifold ( Z , h t ) to properties of the almost Hermitian structure( g, J ). In this note we address the problem of when Σ J is a minimal hypersurfacein the twistor space of a manifold of dimension four. In this dimension, there arethree basic classes in the Gray-Hervella classification - those of Hermitian, almostK¨ahler (symplectic) and K¨ahler manifolds. If ( g, J ) is K¨ahler, Σ J is a totally ge-odesic submanifold, as one can expect. In the case of an Hermitian manifold, weexpress the condition for minimality of Σ J in terms of the Lee form of ( M, g, J ),while for an almost K¨ahler manifold we show that Σ J is minimal if and only if the ⋆ -Ricci tensor of ( M, g, J ) is symmetric. Several example illustrating these resultsare discussed in the last section of the paper.2.
Preliminaries
Let (
M, g ) be an oriented Riemannian manifold of dimension four. The metric g induces a metric on the bundle of two-vectors π : Λ T M → M by the formula g ( v ∧ v , v ∧ v ) = 12 det [ g ( v i , v j )] . The Levi-Civita connection of (
M, g ) determines a connection on the bundle Λ T M ,both denoted by ∇ , and the corresponding curvatures are related by R ( X ∧ Y )( Z ∧ T ) = R ( X, Y ) Z ∧ T + Z ∧ R ( X, Y ) T for X, Y, Z, T ∈ T M . Let us note that we adopt the following definition for thecurvature tensor R : R ( X, Y ) = ∇ [ X,Y ] − [ ∇ X , ∇ Y ].The Hodge star operator defines an endomorphism ∗ of Λ T M with ∗ = Id .Hence we have the decompositionΛ T M = Λ − T M ⊕ Λ T M where Λ ± T M are the subbundles of Λ T M corresponding to the ( ± ∗ .Let ( E , E , E , E ) be a local oriented orthonormal frame of T M . Set s ± = E ∧ E ± E ∧ E , s ± = E ∧ E ± E ∧ E , s ± = E ∧ E ± E ∧ E . (1)Then ( s ± , s ± , s ± ) is a local orthonormal frame of Λ ± T M defining an orientationon Λ ± T M , which does not depend on the choice of the frame ( E , E , E , E ).For every a ∈ Λ T M , define a skew-symmetric endomorphism of T π ( a ) M by g ( K a X, Y ) = 2 g ( a, X ∧ Y ) , X, Y ∈ T π ( a ) M. (2)Note that, denoting by G the standard metric − T race P Q on the space of skew-symmetric endomorphisms, we have G ( K a , K b ) = g ( a, b ) for a, b ∈ Λ T M . If σ ∈ Λ T M is a unit vector, then K σ is a complex structure on the vector space T π ( σ ) M compatible with the metric and the orientation of M . Conversely, the 2-vector σ dual to one half of the K¨ahler 2-form of such a complex structure is aunit vector in Λ T M . Thus the unit sphere subbunlde Z + = Z + ( M ) of Λ T M parametrizes the complex structures on the tangent spaces of M compatible withits metric and orientation. This subbundle is called the twistor space of M .The Levi-Civita connection ∇ of M preserves the bundles Λ ± T M , so it induces ametric connection on these bundles denoted again by ∇ . The horizontal distributionof Λ T M with respect to ∇ is tangent to the twistor space Z + . Thus we have thedecomposition T Z + = H ⊕ V of the tangent bundle of Z + into horizontal andvertical components. The vertical space V τ = { V ∈ T τ Z + : π ∗ V = 0 } at a point τ ∈ Z + is the tangent space to the fibre of Z + through τ . Thus, considering T τ Z + as a subspace of T τ (Λ T M ) (as we shall always do), V τ is the orthogonalcomplement of R τ in Λ T π ( τ ) M . The map V ∋ V τ → K V gives an identificationof the vertical space with the space of skew-symmetric endomorphisms of T π ( τ ) M that anti-commute with K τ . Let s be a local section of Z + such that s ( p ) = τ where p = π ( τ ). Considering s as a section of Λ T M , we have ∇ X s ∈ V τ forevery X ∈ T p M since s has a constant length. Moreover, X hτ = s ∗ X − ∇ X s is thehorizontal lift of X at τ .Denote by × the usual vector cross product on the oriented 3-dimensional vectorspace Λ T p M , p ∈ M , endowed with the metric g . Then it is easy to check that g ( R ( a ) b, c ) = g ( R ( b × c ) , a ) (3) WISTORIAL CONSTRUCTION OF MINIMAL HYPERSURFACES 3 for a ∈ Λ T p M , b, c ∈ Λ T p M . It is also easy to show that for every a, b ∈ Λ T p MK a ◦ K b = − g ( a, b ) Id + K a × b . (4)For every t >
0, define a Riemannian metric h t by h t ( X hσ + V, Y hσ + W ) = g ( X, Y ) + tg ( V, W )for σ ∈ Z + , X, Y ∈ T π ( σ ) M , V, W ∈ V σ .The twistor space Z + admits two natural almost complex structures that arecompatible with the metrics h t . One of them has been introduced by Atiyah,Hitchin and Singer who have proved that it is integrable if and only if the basemanifold is anti-self-dual [1]. The other one, introduced by Eells and Salamon,although never integrable, plays an important role in harmonic maps theory [10].The action of SO (4) on Λ R preserves the decomposition Λ R = Λ R ⊕ Λ − R . Thus, considering S as the unit sphere in Λ R , we have an action of thegroup SO (4) on S . Then, if SO ( M ) denotes the principal bundle of the orientedorthonormal frames on M , the twistor space Z + = Z + ( M ) is the associated bundle SO ( M ) × SO (4) S . It follows from the Vilms theorem (see, for example, [3, Theorem9.59]) that the projection map π : ( Z + , h t ) → ( M, g ) is a Riemannian submersionwith totally geodesic fibres (this can also be proved by a direct computation).Denote by D the Levi-Chivita connection of ( Z + , h t ).Let ( N, x , ..., x ) be a local coordinate system of M and let ( E , ..., E ) be anoriented orthonormal frame of T M on N . If ( s +1 , s +2 , s +3 ) is the local frame of Λ T M define by (1), then e x a = x a ◦ π , y j ( τ ) = g ( τ, ( s + j ◦ π )( τ )), 1 ≤ a ≤
4, 1 ≤ j ≤
3, arelocal coordinates of Λ T M on π − ( N ).The horizontal lift X h on π − ( N ) of a vector field X = X a =1 X a ∂∂x a is given by X h = X a =1 ( X a ◦ π ) ∂∂ e x a − X j,k =1 y j ( g ( ∇ X s j , s k ) ◦ π ) ∂∂y k . (5)Hence [ X h , Y h ] = [ X, Y ] h + X j,k =1 y j ( g ( R ( X ∧ Y ) s j , s k ) ◦ π ) ∂∂y k (6)for every vector fields X, Y on N . Let τ ∈ Z + . Using the standard identification T τ (Λ T p M ) ∼ = Λ T π ( τ ) M we obtain from (6) the well-known formula[ X h , Y h ] τ = [ X, Y ] hτ + R p ( X ∧ Y ) τ, p = π ( τ ) . (7)Then we have the following Lemma 1. ([7]) If X, Y are (local) vector fields on M and V is a vertical vectorfield on Z + , then ( D X h Y h ) τ = ( ∇ X Y ) hτ + 12 R p ( X ∧ Y ) τ, (8) JOHANN DAVIDOV ( D V X h ) τ = H ( D X h V ) τ = − t R p ( τ × V ) X ) hτ (9) where τ ∈ Z + , p = π ( τ ) , and H means ”the horizontal component”. Proof . Identity (8) follows from the Koszul formula for the Levi-Chivita con-nection and (7).Let W be a vertical vector field on Z + . Then h t ( D V X h , W ) = − h t ( X h , D V W ) = 0since the fibres are totally geodesic submanifolds, so D V W is a vertical vector field.Therefore D V X h is a horizontal vector field. Moreover, [ V, X h ] is a vertical vectorfield, hence D V X h = H D X h V . Thus h t ( D V X h , Y h ) = h t ( D X h V, Y h ) = − h t ( V, D X h Y h ) . Now (9) follows from (8) and (3).3.
A hypersurface in Z + determined by an almost Hermitianstructure on M Let ( g, J ) be an almost Hermitian structure on a four-manifold M . Define asection α of Λ T M by g ( α, X ∧ Y ) = 12 g ( JX, Y ) , X, Y ∈ T M.
Thus, at any point of M , α is the dual 2-vector of one half of the K¨ahler 2-formof the almost Hermitian manifold ( M, g, J ). Note also that K α p = J p for every p ∈ M .Consider M with the orientation yielded by the almost complex structure J .Then α is a section of the twistor bundle Z + . This section determines a hypersurfaceof the twistor space defined byΣ J = { σ ∈ Z + : g ( σ, α π ( σ ) ) = 0 } . By (4), the points of Σ J are complex structures on the tangent spaces of M thatare compatible with the metric and the orientation, and anti-commute with J .Clearly, Σ J is the circle bundle of the rank 2 vector bundleΛ = { σ ∈ Λ T M : g ( σ, α π ( σ ) ) = 0 } . As is well-known (and easy to see), the complexification of this bundle is the bundleΛ , ⊕ Λ , where Λ r,s stands for the bundle of ( r + s )-vectors of type ( r, s ) withrespect to J .We shall compute the second fundamental form Π of the hypersurface Σ J in( Z + , h t ).Note that for σ ∈ Σ J T σ Σ J = { E ∈ T σ Z + : g ( V E, α π ( σ ) ) = − g ( σ, ∇ π ∗ E α ) } where V E means ”the vertical component of E ”. Therefore T σ Σ J = { X hσ − g ( σ, ∇ X α ) α π ( σ ) : X ∈ T π ( σ ) M } ⊕ R ( α π ( σ ) × σ ) . Given τ ∈ Z and X ∈ T π ( τ ) M , define a vertical vector of Z + at τ by X vτ = − g ( τ, ∇ X α ) α π ( τ ) + g ( τ, α π ( τ ) ) ∇ X α. Set b X τ = X hτ + X vτ . WISTORIAL CONSTRUCTION OF MINIMAL HYPERSURFACES 5
Thus every (local) vector field X on M , gives rise to a vector field b X on Z + tangentto Σ J .Let ρ ( τ ) = g ( τ, α π ( τ ) ), τ ∈ Z + , be the defining function of Σ J and let grad ρ be the gradient vector field of the function ρ with respect to the metric h t . Fix apoint τ ∈ Z + and take a section s of Z + such that s π ( τ ) = τ , ∇ s | π ( τ ) = 0. Then,for X ∈ T π ( τ ) M , h t ( X hτ , grad ρ ) = s ∗ ( X )( ρ ) = X ( g ( s, α )) = g ( τ, ∇ X α ) . (10)Moreover, if V ∈ V τ , h t ( V, grad ρ ) = V ( X k =1 y k ( g ( s k , α ) ◦ π )) = X k =1 V ( y k ) g ( s k , α ) π ( τ ) = g ( V, α π ( τ ) ) . (11) Lemma 2. If σ ∈ Σ J and X, Y ∈ T π ( σ ) M , then h t (Π( b X, b Y ) , grad ρ ) σ = t g ( σ, ∇ X α ) g ( σ, ∇ R ( σ × α π ( σ ) ) Y α )+ g ( σ, ∇ Y α ) g ( σ, ∇ R ( σ × α π ( σ ) ) X α )] − g ( σ, ∇ XY α ) − g ( σ, ∇ Y X α ) where ∇ XY α = ∇ X ∇ Y α − ∇ ∇ X Y α is the second covariant derivative of α . Proof . Extend X and Y to vector fields in a neighbourhood of the point p = π ( σ ). It follows from (8), (10) and (11) that h t ( D X h Y h , grad ρ ) σ = g ( ∇ ∇ X Y α, σ ) + 12 g ( R ( X ∧ Y ) σ, α p ) . (12)Identities (9) and (10) imply h t ( D X v Y h , grad ρ ) σ = t g ( σ, ∇ X α ) g ( σ, ∇ R ( σ × α p ) Y α ) . (13)Next, note that h t ( D X h Y v , grad ρ ) = h t ([ X h , Y v ] , grad ρ ) + h t ( D Y v X h , grad ρ ) . Take an oriented orthonormal frame ( E , ..., E ) of M near p such that ∇ E a | p = 0, a = 1 , ...,
4. Then ∇ s + i | p = 0, i = 1 , ,
3, which implies X hσ = X a =1 X a ( p ) ∂∂ e x a ( σ ) , [ X h , ∂∂y i ] σ = 0 , i = 1 , , . (14)We have Y v = X j,k =1 y k ( g ( s + k , α ) g ( ∇ Y α, s + j ) − g ( s + j , α ) g ( ∇ Y α, s + k )) ◦ π ∂∂y j . (15)It follows from (14) and (15) that[ X h , Y v ] σ = g ( σ, ∇ X α ) ∇ Y α − g ( σ, ∇ Y α ) ∇ X α − g ( σ, ∇ X ∇ Y α ) α p . Hence, by (11), h t ([ X h , Y v ] , grad ρ ) σ = − g ( σ, ∇ X ∇ Y α ) . JOHANN DAVIDOV
Thus we have h t ( D X h Y v , grad ρ ) σ = − g ( σ, ∇ X ∇ Y α ) + t g ( σ, ∇ Y α ) g ( σ, ∇ R ( σ × α p ) X α ) . (16)The fibres of of Z + are totally geodesic submanifolds, hence ( D X v Y v ) σ is thestandard covariant derivative on the unit sphere in the vector space Λ T p M . Itfollows from (15) that( D X v Y v ) σ = g ( X vσ , α p )[ ∇ Y α − g ( ∇ Y α, σ ) σ ] − g ( X vσ , ∇ Y α )[ α p − g ( α p , σ ) σ ] = − g ( σ, ∇ X α )[ ∇ Y α − g ( ∇ Y α, σ ) σ ] . Hence h t ( D X v Y v , grad ρ ) σ = 0 . (17)Now the lemma follows from identities (12), (13), (16), and (17).If σ ∈ Σ J , the vertical part of T σ Σ J is R ( α π ( σ ) × σ ). Define a vertical vectorfield ξ on Z + tangent to Σ J setting ξ τ = α π ( τ ) × τ, τ ∈ Z + . Lemma 3. If σ ∈ Σ J and X ∈ T π ( σ ) M , then h t (Π( ξ, b X ) , grad ρ ) σ = − g ( ξ σ , ∇ X α ) − t g ( σ, ∇ R ( α π ( σ ) ) X α ) .h t (Π( ξ, ξ ) , grad ρ ) σ = 0 . Proof . Identity (9) implies h t ( D ξ X h , grad ρ ) σ = − t g ( σ, ∇ R ( σ × ξ σ ) X α ) . (18)A simple computation gives( D ξ X v ) σ = − g ( ξ σ , ∇ X α ) α π ( σ ) , ( D ξ ξ ) σ = 0 . Hence h t ( D ξ X v , grad ρ ) σ = − g ( ξ σ , ∇ X α ) , h t ( D ξ ξ, grad ρ ) σ = 0 . (19)Thus the result follows from (18) and (19). Proposition 1.
Let σ ∈ Σ J and E, F ∈ T σ Σ J . Set X = π ∗ E , Y = π ∗ F , V = V E , W = V F . Then h t (Π( E, F ) , grad ρ ) σ = t g ( σ, ∇ X α ) g ( σ, ∇ R ( σ × α π ( σ ) ) Y α ) + t g ( σ, ∇ Y α ) g ( σ, ∇ R ( σ × α π ( σ ) ) X α ) − g ( σ, ∇ XY α ) − g ( σ, ∇ Y X α )+ t g ( α π ( σ ) × V, ∇ R ( α π ( σ ) ) Y α ) + t g ( α π ( σ ) × W, ∇ R ( α π ( σ ) ) X α ) − g ( V, ∇ Y α ) − g ( W, ∇ X α ) . Proof . This follows from Lemmas 2 and 3 taking into account that E = b X σ + g ( V, ξ σ ) ξ σ , F = b Y σ + g ( W, ξ σ ) ξ σ . Corollary 1. If ( M, g, J ) is K¨ahler, Σ J is a totally geodesic submanifold of Z + . WISTORIAL CONSTRUCTION OF MINIMAL HYPERSURFACES 7 Minimality of the hypersurface Σ J Let Ω(
X, Y ) = g ( JX, Y ) be the fundamental 2-form of the almost Hermitianmanifold (
M, g, J ). Denote by N the Nijenhuis tensor of J , N ( Y, Z ) = − [ Y, Z ] +[
JY, JZ ] − J [ Y, JZ ] − J [ JY, Z ]. It is well-known (and easy to check) that2 g (( ∇ X J )( Y ) , Z ) = d Ω( X, Y, Z ) − d Ω( X, JY, JZ ) + g ( N ( Y, Z ) , JX ) . (20)4.1. The case of integrable J . Suppose that the almost complex structure J isintegrable. Note that the integrability condition for J is equivalent to ( ∇ X J )( Y ) =( ∇ JX J )( JY ), X, Y ∈ T M [12, Corollary 4.2]. Let B be the vector field on M dualto the Lee form θ = − δ Ω ◦ J with respect to the metric g . Then (20) and theidentity d Ω = Ω ∧ θ imply the following well-known formula2( ∇ X J )( Y ) = g ( JX, Y ) B − g ( B, Y ) JX + g ( X, Y ) JB − g ( JB, Y ) X. (21)We have g ( ∇ X α, Y ∧ Z ) = 12 g (( ∇ X J )( Y ) , Z ) and it follows that ∇ X α = 12 ( JX ∧ B + X ∧ JB ) . (22)The latter identity implies ∇ XY α = 12 [( ∇ X J )( Y ) ∧ B + Y ∧ ( ∇ X J )( B ) + JY ∧ ∇ X B + Y ∧ J ∇ X B ] . (23)Let σ ∈ Σ J and X, Y ∈ T π ( σ ) M . Then a simple computation using identities (2),(4), (21) - (23) gives g ( σ, ∇ X α ) = 12 g ( X, K ξ σ B ) , g ( σ, ∇ XY α ) = − g ( K ξ σ B ∧ B, X ∧ Y + JX ∧ JY ) − g ( JX, B ) g ( JY, K ξ σ B )+ 12 || B || g ( X, K ξ σ Y ) − g ( ∇ X B, K ξ σ Y )where, as above, ξ σ = α π ( σ ) × σ . Moreover, if V ∈ T σ Σ J is a vertical vector, g ( α π ( σ ) × V, ∇ X α ) = − g ( V, ξ σ ) g ( σ, JX ∧ B + X ∧ JB ) = − g ( V, ξ σ ) g ( X, K ξ σ B ) ,g ( V, ∇ X α ) = − g ( V, ξ σ ) g ( X, K σ B ) . Now Proposition 1 can be rewritten as
Proposition 2.
Suppose that the almost complex structure J is integrable. Let σ ∈ Σ J and E, F ∈ T σ Σ J . Set X = π ∗ E , Y = π ∗ F , V = V E , W = V F and JOHANN DAVIDOV ξ σ = α π ( σ ) × σ . Then h t (Π( E, F ) , grad ρ ) σ = − t g ( X, K ξ σ B ) g ( Y, R ( ξ σ ) K ξ σ B ) − t g ( Y, K ξ σ B ) g ( X, R ( ξ σ ) K ξ σ B )+ 18 g ( JX, B ) g ( JY, K ξ σ B ) + 18 g ( JY, B ) g ( JX, K ξ σ B )+ 14 g ( ∇ X B, K ξ σ Y ) + 14 g ( ∇ Y B, K ξ σ X ) − g ( V, ξ σ ) g ( R ( α π ( σ ) ) Y, K ξ σ B ) − g ( W, ξ σ ) g ( R ( α π ( σ ) ) X, K ξ σ B )+ 14 g ( V, ξ σ ) g ( X, K σ B ) + 14 g ( W, ξ σ ) g ( Y, K σ B ) Corollary 2.
Let σ ∈ Σ J . Then h t ( T race Π , grad ρ ) σ = 12 ( dθ + θ ∧ d ln q t || θ π ( σ ) || )( α π ( σ ) × σ ) . Proof . Set p = π ( σ ). Suppose first that B p = 0. Then E = || B p || − B p , E = K α p E , E = K σ E , E = K ξ σ E form an oriented orthonormal basis of T p M such that α p = ( s ) + p , σ = ( s ) + p , ξ σ = ( s ) + p where s +1 , s +2 , s +3 is the basis ofΛ T p M defined by means of E , ..., E via (1). We have g ( σ, ∇ X α ) = 12 g ( X, K ξ σ B ) = 12 || B p || g ( X, E ) , X ∈ T p M. Hence d ( E i ) σ = ( E i ) hσ for i = 1 , , d ( E ) σ = ( E ) hσ − || B p || α p . Thus d ( E i ) σ , i = 1 , ,
3, (1 + t || B p || ) − d ( E ) σ , 1 √ t ξ σ constitute an orthonormal basis of T σ Σ J .Note that g ( E , R ( ξ σ ) K ξ σ B ) = || B p || − g ( K ξ σ B, R ( ξ σ ) K ξ σ B ) = 0and g ( JE , B ) = || B p || − g ( K α p ◦ K ξ σ B, B ) = −|| B p || − g ( K σ B, B ) = 0 . Then, by Proposition 2, h t ( T race Π , grad ρ ) σ = − g ( K ξ σ B, R ( ξ σ ) K ξ σ B ) + 14 g ( B, K ξ σ B )+ 12 X i =1 g ( ∇ E i B, K ξ σ E i ) + 12 (1 + t || B p || ) − g ( ∇ E B, K ξ σ E )= 12 X j =1 g ( ∇ E j B, K ξ σ E j ) + t || B p || t || B p || g ( ∇ E B, B )= 12 dθ ( E ∧ E + E ∧ E ) + t
16 + 4 t || B p || ( θ ∧ d || θ || )( E ∧ E + E ∧ E )If B p = 0, then ∇ X α = 0 for every X ∈ T p M by (22). Taking a unit vector E ∈ T p M we set E = K α p E , E = K σ E , E = K ξ σ E . Then d ( E j ) σ =( E j ) hσ , j = 1 , ...,
4, 1 √ t ξ σ constitute an orthonormal basis of T σ Σ J . It follows from WISTORIAL CONSTRUCTION OF MINIMAL HYPERSURFACES 9
Proposition 2 that h t ( T race Π , grad ρ ) σ = 12 X j =1 g ( ∇ E j B, K ξ σ E j ) = 12 dθ ( E ∧ E + E ∧ E ) . Proposition 3. If J is integrable, the hypersurface Σ J is a minimal submanifoldof ( Z + , h t ) if and only if the -form d θ p t || θ || is of type (1 , with respect to J . Proof . The condition that Σ J is a minimal submanifold means that h t ( T race Π , grad ρ ) = 0 on Σ J .Let p ∈ M and take an orthonormal basis of T p M of the form E , E = JE , E , E = JE . Then α p = ( s ) + p , so ( s ) + p , ( s ) + p ∈ Σ J .It is easy to check that, for every a ∈ Λ T p M and b ∈ Λ − T p M , the endomor-phisms K a and K b of T p M commute. It follows that, for every X, Y ∈ T p M , the2-vector X ∧ Y − JX ∧ JY is orthogonal to Λ − T p M , therefore it lies in Λ T p M .Moreover, X ∧ Y − JX ∧ JY is orthogonal to α p , hence is a linear combination of( s ) + p = − α p × ( s ) + p and ( s ) + p = α p × ( s ) + p . Thus if h t ( T race Π , grad ρ ) = 0 onΣ J , ( dθ + θ ∧ d ln p t || θ || )( X ∧ Y − JX ∧ JY ) = 0 . Conversely, if this identity holds, then h t ( T race Π , grad ρ ) = 0 at the points ( s ) + p and ( s ) + p of Σ J . For every σ ∈ Σ J with π ( σ ) = p , the 2-vector α p × σ is a linearcombination of ( s ) + p and ( s ) + p , hence h t ( T race Π , grad ρ ) = 0 on Σ J .Thus, Σ J is minimal if and only if the form dθ + θ ∧ d ln p t || θ || is oftype (1 , d θ p t || θ || is of type (1 , The case of symplectic J . Now suppose that d Ω = 0. Then, by (20), g (( ∇ X J )( Y ) , Z ) = 12 g ( N ( Y, Z ) , JX ) . The Nijenhuis tensor N ( Y, Z ) is skew-symmetric, so it induces a linear map Λ T M → T M which we denote again by N . The identity g ( ∇ X α, Y ∧ Z ) = 14 g ( N ( Y, Z ) , JX ) (24)implies that, for every a ∈ Λ T M and X ∈ T π ( a ) M , g ( ∇ X α, a ) = 14 g ( N ( a ) , JX ) . In particular, if V ∈ V σ , then g ( ∇ X α, V ) = 14 g ( V, ξ σ ) g ( N ( ξ σ ) , JX ) . Proposition 1 implies the following.
Proposition 4.
Suppose that d Ω = 0 . Let σ ∈ Σ J and E, F ∈ T σ Σ J . Set X = π ∗ E , Y = π ∗ F , V = V E , W = V F and ξ σ = α π ( σ ) × σ . Then h t (Π( E, F ) , grad ρ ) σ = t g ( JN ( σ ) , X ) g ( JN ( σ ) , R ( ξ σ ) Y ) + t g ( JN ( σ ) , Y ) g ( JN ( σ ) , R ( ξ σ ) X ) − g ( σ, ∇ XY α ) − g ( σ, ∇ Y X α )+ t g ( N ( α π ( σ ) × V ) , JR ( α π ( σ ) ) Y ) + t g ( N ( α π ( σ ) × W ) , JR ( α π ( σ ) ) X ) − g ( V, ξ σ ) g ( N ( ξ σ ) , JX ) − g ( W, ξ σ ) g ( N ( ξ σ ) , JY ) . Corollary 3.
Let σ ∈ Σ J . Then h t ( T race Π , grad ρ ) σ = − g ( trace ∇ α, σ ) . Proof . Suppose first that N ( σ ) = 0. Take an orthonormal basis of T π ( σ ) M ofthe form E , E = JE , E = || N ( σ ) || − N ( σ ), E = JE . Then, by (24), [ ( E k ) σ = ( E k ) hσ + 14 || N ( σ ) || g ( E , E k ) α π ( σ ) ,k = 1 , ...,
4. Thus d ( E i ) σ , i = 1 , ,
3, (1 + t || N ( σ ) || ) − d ( E ) σ , 1 √ t ξ σ form anorthonormal basis of T σ Σ J . Note also that g ( JN ( σ ) , R ( ξ σ ) E ) = || N ( σ ) || g ( E , R ( ξ σ ) E ) = 0 . Then Proposition 4 implies h t ( T race Π , grad ρ ) σ = − t g ( JN ( σ ) , R ( ξ σ ) JN ( σ )) − g ( trace ∇ α, σ )= − g ( trace ∇ α, σ ) . If N ( σ ) = 0, then, in view of (24), g ( ∇ X α, σ ) = 0. Thus [ ( E k ) σ = ( E k ) hσ for anyorthonormal basis E k of T π ( σ ) M , k = 1 , ...,
4, and the result is a direct consequenceof Proposition 4.Denote by ρ ∗ the ∗ -Ricci tensor of the almost Hermitian manifold ( M, g, J ).Recall that it is defined as ρ ∗ ( X, Y ) = trace { Z → R ( JZ, X ) JY } . Note that ρ ∗ ( JX, JY ) = ρ ∗ ( Y, X ) , (25)in particular ρ ∗ ( X, JX ) = 0.
Proposition 5. If d Ω = 0 , then the hypersurface Σ J is a minimal submanifold of ( Z + , h t ) if and only if the tensor ρ ∗ is symmetric. Proof . The form Ω is harmonic since d Ω = 0 and ∗ Ω = Ω. Then, by Corollary 3and the Weitzenb¨ock formula, Σ J is minimal if and only if, for every 2-form τ ∈ Λ T ∗ M orthogonal to Ω, g ( S (Ω) , τ ) = 0 where S (Ω)( X, Y ) =
T race { Z → ( R ( Z, Y )Ω)(
Z, X ) − ( R ( Z, X )Ω)(
Z, Y ) } (see, for example, [9]). We have( R ( Z, Y )Ω)(
Z, X ) = − Ω( R ( Z, Y ) Z, X ) − Ω( Z, R ( Z, Y ) X )= g ( R ( Z, Y ) Z, JX ) + g ( R ( Z, X ) Y, JZ ) . WISTORIAL CONSTRUCTION OF MINIMAL HYPERSURFACES 11
Hence S (Ω)( X, Y ) =
Ricci ( Y, JX ) − Ricci ( X, JY ) + 2 ρ ∗ ( X, JY ) . By (25), in order to show that ρ ∗ ( X, Y ) = ρ ∗ ( Y, X ) for every
X, Y , it is enoughto check that ρ ∗ ( X, Y ) = ρ ∗ ( Y, X ) for all unit vectors
X, Y ∈ T M with g ( X, Y ) = g ( X, JY ) = 0. If X , Y are such vectors, E = X , E = JE , E = Y , E = JY isan orthonormal basis and the condition g ( S (Ω) , τ ) = 0 is equivalent to S (Ω)( E , E ) + S (Ω)( E , E ) = 0 , S (Ω)( E , E ) + S (Ω)( E , E ) = 0 . (26)These identities are equivalent to ρ ∗ ( E , E ) = ρ ∗ ( E , E ) and ρ ∗ ( E , E ) = ρ ∗ ( E , E )where ρ ∗ ( E , E ) = ρ ∗ ( JE , JE ) = ρ ∗ ( E , E ). Taking into account (25) we seethat (26) is equivalent to ρ ∗ ( X, Y ) = ρ ∗ ( Y, X ).5.
Examples
Generalized Hopf surfaces.
Clearly, if M is locally conformally K¨ahler( dθ = 0) and the Lee form θ has constant length, the hypersurface Σ J is minimal.We have || θ || ≡ const on every homogeneous locally conformally K¨ahler manifold.Also, if θ is parallel, then dθ = 0 and || θ || ≡ const . Recall that a Hermitian surfacewith parallel Lee form is called a generalized Hopf surface [18] (or a Vaisman surface[8]); we refer to [8, 18] for basic properties and examples of such surfaces. Theproduct of a Sasakian 3-manifold with R or S admits a structure of generalizedHopf surface in a natural way. Conversely, every such a surface locally is the productof a Sasakian manifold and R [18] (cf. also [11]). A global structure theorem forcompact generalized Hopf manifolds is obtained in [16].As it is shown in [17], certain Inoue surfaces admit locally conformally K¨ahlerstructures with || θ || ≡ const and non-parallel Lee form θ .Fix two complex numbers α and β such that | α | ≥ | β | >
1. Let Γ α,β be the groupof transformation of C \ { } generated by the transformation ( u, v ) → ( αu, βv ).Then, by a result of [11], the quotient M α,β = ( C \ { } ) / Γ α,β admits a structure ofa generalized Hopf surface. Note that M α,β (as any primary Hopf complex surface)is diffeomorphic to S × S . The sypersurface Σ J in the twistor space of S × S We shall consider the Hopf surface S × S with its standard complex structure J and the product metric.According to [5, Example 5], we have Z + ( S × S ) ∼ = { [ z , z , z , z ] ∈ CP : | z | + | z | = | z | + | z |} × S . In order to give an explicit description of this isomorphism, we first recall thatthe twistor space of an odd-dimensional oriented Riemannian manifold (
M, g ) isthe bundle C + ( M ) over M whose fibre at a point p ∈ M consists of all (linear)contact structures on the tangent space T p M compatible with the metric and theorientation, i.e. pairs ( ϕ, ξ ) of endomorphism ϕ of T p M and a unit vector ξ ∈ T p M such that ϕ X = − X + g ( X, ξ ) ξ , g ( ϕX, ϕY ) = g ( X, Y ) − g ( X, ξ ) g ( Y, ξ ) for
X, Y ∈ T p M , and the orientation of T p M is induced by the orthogonal decomposition T p M = Im ϕ ⊕ R ξ where the vector space Im ϕ is oriented by means of the complexstructure ϕ | Im ϕ on it. We refer to [5, 6] and the references therein for moreinformation about the twistor spaces of odd-dimensional manifolds. The twistor space C + ( M ) admits a 1-parameter family of Riemannian metrics h ct defined in away similar to the definition of the metrics h t on the twistor space Z + .As is well-known, given ( ϕ, ξ ) and a ∈ S , we can define a complex structure I on T p M × T a S in the following way. Denote by ∂∂t the vector field on S determined by the local coordinate e it → t . Then set I = ϕ on Im ϕ , Iξ = ( ∂∂t )( a ), I ( ∂∂t )( a ) = − ξ . The complex structure I is compatible with the product metric of M × S and its orientation, S (as well as any other sphere) being oriented by theinward normal vector field. In this way we have a map F : C + ( M ) × S → Z + ( M × S ) . Endow C + ( M ) × S with the product metric. It is a simple observation that the map F is a bundle isomorphism preserving the metrics (and having other nice properties)[5, Example 4]. Now we apply this observation to the case when M = S and shalldefine an embedding of C + ( S ) into CP as in [6, Examples 2 and 3].Denote the standard basis of R by a , ..., a and consider R and R as thesubspaces span { a , a , a } and span { a , ..., a } . Let ( ϕ, ξ ) ∈ C + ( S ) with ϕ ∈ End ( T p S ) and ξ ∈ T p M , p ∈ S . Then we define a complex structure J on R bymeans of the orthogonal decomposition R = Im ϕ ⊕ R ξ ⊕ R {− p } ⊕ R a ⊕ R a setting J = ϕ on Im ϕ , Jξ = − a , Jp = − a , Ja = ξ , Ja = p . In this way weobtain an embedding κ of C + ( S ) into the space J + ( R ) of complex structures on R compatible with the metric and the orientation. The tangent space of J + ( R )at any point I consists of skew-symmetric endomorphisms Q of R anti-commutingwith I . Denote by G the standard metric − T race P Q on the space of skew-symmetric endomorphisms. Then κ ∗ G = h c / [6, Example 2]. It is well-knownthat J + ( R ) and CP are both isomorphic to the twistor space of S (see, forexample, [20]), so J + ( R ) ∼ = CP ; for a direct proof see [2, 19]). We shall makeuse of the biholomorphism that sends a point [ z , z , z , z ] ∈ CP to the complexstructure J of R defined as follows: Let a , ..., a be the standard basis of R andset A k = 1 √ a k − − ia k ) , k = 1 , ,
3. Then the structure J is given by − i | z | JA = ( | z | −| z | −| z | + | z | ) A +2 z z A +2 z z A +2 z z A − z z A − i | z | JA = 2 z z A +( −| z + | z | −| z | + | z | ) A +2 z z A − z z A +2 z z A − i | z | JA = 2 z z A +2 z z A +( −| z | −| z | + | z | + | z | ) A +2 z z A − z z A . where z = ( z , z , z , z ).For every p ∈ S , denote by × the vector cross-product on the oriented 3-dimensional Euclidean space T p S . If ( ϕ, ξ ) is a linear contact structure on T p M compatible with the metric and the orientation, then ϕ ( v ) = ξ × v, v ∈ T p S . In particular, ( ϕ, ξ ) is uniquely determined by ξ . Define an oriented orthonormalglobal frame of the bundle T S by ξ ( p ) = ( − p , p , − p , p ) , ξ ( p ) = ( − p , p , p , − p ) , ξ ( p ) = ( − p , − p , p , p ) , for ( p , p , p , p ) ∈ S . WISTORIAL CONSTRUCTION OF MINIMAL HYPERSURFACES 13
Set ϕ ( v ) = ξ ( p ) × v , v ∈ T p S . Then the standard complex structure J of S × S corresponds to the section ( ϕ , ξ ) × ∂∂t of C + ( S ) × S under the isomorphism F : C + ( S ) × S → Z + ( S × S ). We note also that if J ′ , J ′′ corresponds to( ϕ ′ , ξ ′ ), ( ϕ ′′ , ξ ′′ ) under F , then G ( J ′ , J ′′ ) = g ( ξ ′ , ξ ′′ ). In particular, J ′ and J ′′ areorthogonal if and only if ξ ′ and ξ ′′ are so. Let ( ϕ, ξ ) ∈ C + ( S ) and ξ ⊥ ξ ( p ),thus ξ = λ ξ ( p ) + λ ξ ( p ) where λ + λ = 1. Then the point [ z , ..., z ] ∈ CP corresponding to ( ϕ, ξ ) under the embedding C + ( S ) ֒ → J + R ∼ = CP is given by z = 12 [ − ( p + ip ) − ( λ − iλ )( p − ip )] , z = 12 [( λ − iλ )( p − ip ) − ( p + ip ) ,z = 12 , z = −
12 ( λ − iλ ) . In particular, we have 4 | z | = 4 | z | = | z | . Conversely, let [ z , ..., z ] ∈ CP bea point for which 4 | z | = | z | and 4 | z | = | z | . Let p , ..., p , λ , λ be the realnumbers determined by the equations p + ip = − z ¯ z + ¯ z z ) | z | , p + ip = 2(¯ z z − z ¯ z ) | z | , λ + iλ = − z ¯ z | z | . Then p = ( p , ..., p ) ∈ S , λ + λ = 1 and [ z , ..., z ] corresponds under theembedding C + ( S ) ֒ → J + R ∼ = CP to ( ϕ, ξ ) determined by ξ = λ ξ ( p ) + λ ξ ( p ).It follows thatΣ J ∼ = { [ z , z , z , z ] ∈ CP : 4 | z | = 4 | z | = | z | } × S . Kodaira surfaces.
Recall that every primary Kodaira surface M can be ob-tained in the following way [14, p.787]. Let ϕ k ( z, w ) be the affine transformationsof C given by ϕ k ( z, w ) = ( z + a k , w + a k z + b k ) , where a k , b k , k = 1 , , ,
4, are complex numbers such that a = a = 0 , Im ( a a ) = mb = 0 , b = 0for some integer m >
0. They generate a group G of transformations acting freelyand properly discontinuously on C , and M is the quotient space C /G .It is well-known that M can also be describe as the quotient of C endowed witha group structure by a discrete subgroup Γ. The multiplication on C is defined by( a, b ) . ( z, w ) = ( z + a, w + az + b ) , ( a, b ) , ( z, w ) ∈ C , and Γ is the subgroup generated by ( a k , b k ), k = 1 , ..., M as the quotient of the group C by the discretesubgroup Γ. Every left-invariant object on C descends to a globally defined objecton M and both of them will be denoted by the same symbol.We identify C with R by ( z = x + iy, w = u + iv ) → ( x, y, u, v ) and set A = ∂∂x − x ∂∂u + y ∂∂v , A = ∂∂y − y ∂∂u − x ∂∂v , A = ∂∂u , A = ∂∂v . These fields form a basis for the space of left-invariant vector fields on C . We notethat the Lie brackets of the vector fields A , ..., A are[ A , A ] = − A , [ A i , A j ] = 0 for all other i, j, i < j. It follows that the group C defined above is solvable. Denote by g the left-invariant Riemannian metric on M for which the basis A , ..., A is orthonormal.We shall consider almost complex structures J on M compatible with the metric g obtained from left-invariant almost complex structures on C . Note that by [13]every complex structure on M is induced by a left-invariant complex structure. I . If J is a left-invariant almost complex structure compatible with g , we have JA i = P j =1 a ij A j where a ij are constants with a ij = − a ji . Let N be the Nijenhuistensor of J . Computing N ( A i , A j ) in terms of a ij , one can see [15] that J isintegrable if and only if JA = ε A , JA = ε A , ε , ε = ± . Since we are dealing with the complex structures orthogonal to J , it is enough toconsider the two structures J ε defined by JA = εA , JA = A , ε = ± . Endow M with the orientation induced by J ε . Then Λ M admits a global or-thonormal frame defined by s ε = εA ∧ A + A ∧ A , s ε = A ∧ A + εA ∧ A , s ε = A ∧ A + εA ∧ A . Hence we have a natural diffeomorphism F ε : Z + ( M ) ∼ = M × S , X k =1 x k s εk ( p ) → ( p, x , x , x ) , under which Σ J ε ∼ = { ( p, x ) ∈ M × S : x = 0 } . In order to find an explicit formula for the metrics h t we compute the covariantderivatives of s ε , s ε , s ε with respect to the Levi-Civita connection ∇ of g . Thenon-zero covariant derivatives ∇ A i A j are ∇ A A = −∇ A A = − A , ∇ A A = ∇ A A = A , ∇ A A = ∇ A A = − A . Then ∇ A s ε = −∇ A s ε = − εs ε , ε ∇ A s ε = −∇ A s ε = s ε ; ∇ A s ε = − ε ∇ A s ε = s ε . and all other covariant derivatives ∇ A i s εk are zero. It follows that F ε ∗ sends thehorizontal lifts A h , ..., A h at a point σ = P k =1 x k s εk ( p ) ∈ Z + ( M ) to the vectors A + ε ( − x , , x ) , A + ( x , − x , , A , A + ε (0 , x , − x ) . For x = ( x , x , x ) ∈ S , set u ε ( x ) = ε ( − x , , x ) , u ε ( x ) = ( x , − x , , u ε ( x ) = 0 , u ε ( x ) = ε (0 , x , − x ) . These are tangent vectors to S at the point x . Denote by h εt the pushforward ofthe metric h t by F ε Then, if
X, Y ∈ T p M and P, Q ∈ T x S , h εt ( X + P, Y + Q ) = g ( X, Y ) + t < P − X i =1 g ( X, A i ) u εi ( x ) , Q − X j =1 g ( Y, A j ) u εj ( x ) > (27)where < ., . > is the standard metric of R .Let θ ε be the Lee form of the Hermitian manifold ( M, g, J ε ). We have θ ε ( X ) = − εg ( X, A ) which implies ∇ θ ε = 0. Therefore, by Proposition 3, the hypersurface WISTORIAL CONSTRUCTION OF MINIMAL HYPERSURFACES 15 { ( p, x ) ∈ M × S : x = 0 } in M × S is minimal with respect to the metrics h εt given by (27). II . Suppose again that J is an almost complex structure on M obtained froma left-invariant almost complex structure on G and compatible with the metric g . Denote the fundamental 2-form of the almost Hermitian structure ( g, J ) byΩ. Set JA i = P j =1 a ij A j . The basis dual to A , ..., A is α = dx , α = dy , α = xdx + ydy + du , α = − ydx + xdy + dv . We have dα = dα = dα = 0, dα = 2 dx ∧ dy . Hence d Ω = d P i
Proc. R. Soc. London Ser. A , (1978) 435-461.[2] E. Abbena, S. Garbiero, S. Salamon, Hermitian geometry on the Iwasawa manifolds, Boll.Unione Mat. Ital. , VII, ser.B (2), Suppl. (1997) 231-249.[3] A. Besse , Einstein manifolds , Classics in Mathematics (Springer-Verlag, Berlin, 2008).[4] C. Borcea, Moduli for Kodaira surfaces,
Compo. Math. (1984) 373-380.[5] J. Davidov, Twistorial examples of almost contact metric manifolds, Houston J. Math. (2002) 711-740.[6] J. Davidov, Almost contact metric structures and twistor spaces, Houston J. Math. (2003) 639-674.[7] J. Davidov, O. Mushkarov, On the Riemannian curvature of a twistor space, Acta Math.Hungar. (1991) 319-332.[8] S. Dragomir, L. Ornea, Locally conformally K¨ahler geometry , Progress in Mathematics,Vol. 155 (Birckh¨auser, Boston, MA, 1998).[9] J. Eeells,L. Lemaire,
Selected topics in harmonic maps , CBMS Regional Conference Seriesin Mathematics, Vol. 50 (American Mathematical Society, Providernce, RI, 1983).[10] J. Eeells, S. Salamon, Twistorial constructions of harmonic maps of surfaces into four-manifolds,
Ann. Scuola Norm. Sup. Pisa (1985) 589-640.[11] P. Gauduchon, L. Ornea Locally conformally K¨ahler metrics on Hopf surfaces, Ann. Inst.Fourier (Grenoble) (1998) 1107-1127.[12] A. Gray, Minimal varieties and almost Hermitian manifolds, Michigan Math. J. (1965)273-287.[13] K. Hasegawa, Complex and K¨ahler structures on compact solvmanifolds, J. SymplecticGeom. (2005) 749-767.[14] K. Kodaira, On the structure of compact complex analytic surfaces I. Amer. J. Math. (1964) 751-798.[15] O. Muˇskarov, Two remarks on Thurston’s example, in Complex Analysis and Applications’85 (Publ. House Bulgar. Acad. Sci., Sofia, 1986), pp. 461-468.[16] L. Ornea, M. Verbitsky, Structure theorem for compact Vaisman manifolds,
Math. Res.Lett. (2003) 799-805.[17] F. Tricerri, Some examples of locally conformally K¨ahler manifolds, Rend. Sem. Mat. Univ.Polytec. Torino (1982) 81-92.[18] I. Vaisman, Generalized Hopf manifolds, Geom.Dedicata (1982) 231-255.[19] H. Wang, Twistor spaces over 6-dimensional Riemannian manifolds, Illinois J. Math. (1987) 274-311. WISTORIAL CONSTRUCTION OF MINIMAL HYPERSURFACES 17 [20] T. J. Willmore,
Riemannian geometry (Oxford University Press, 1993).
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad.G.Bonchev Str. Bl.8, 1113 Sofia, Bulgaria and, ”L. Karavelov” Civil EngineeringHigher School, 1373 Sofia.
E-mail address ::