On harmonic and pseudoharmonic maps from strictly pseudoconvex CR manifolds
aa r X i v : . [ m a t h . DG ] M a y On Harmonic and Pseudoharmonic Maps from StrictlyPseudoconvex CR Manifolds
Tian Chong Yuxin Dong Yibin Ren Guilin Yang
Abstract. In this paper, we give some rigidity results for both harmonic and pseu-doharmonic maps from CR manifolds into Riemannian manifolds or K¨ahler manifolds.Some basicity, pluriharmonicity and Siu-Sampson type results are established for bothharmonic maps and pseudoharmonic maps.
In 1980, Siu [22] studied the strong rigidity of compact K¨ahler manifolds by using thetheory of harmonic maps. The basic discovery by Siu was a new Bochner-type formula forharmonic maps between K¨ahler manifolds, which does not involve the Ricci curvature tensorof the domains. Using the modified Bochner formula, he proved that any harmonic mapsfrom a compact K¨ahler manifold to a K¨ahler manifold with strongly semi-negative curvatureare actually pluriharmonic and some curvature terms of the pull-back complexifed tangentbundles vanish. When the target manifolds are K¨ahler manifolds with strongly negativecurvature or compact quotients of irreducible bounded symmetric domains, the vanishingcurvature terms, under the assumption of sufficiently high rank, force the maps to be eitherholomorphic or anti-holomorphic. Later, Sampson [21] showed that any harmonic maps fromcompact K¨ahler manifolds into Riemannian manifolds with nonpositive Hermitian curvatureare also pluriharmonic, which generalized the pluriharmonicity result of Siu to more generaltargets. Pluriharmonic maps, holomorphic maps and Siu-Sampson type results have manyimportant applications in geometry and topology of K¨ahler manifolds. The readers arerefered to [25] for details.In 2002, Petit [17] established some rigidity results for harmonic maps from strictly pseu-doconvex CR manifolds to K¨ahler manifolds and Riemannian manifolds by using tools ofSpinorial geometry. First, he proved that any harmonic map from a compact Sasakian man-ifold to a Riemannian manifold with nonpositive sectional curvature is trivial on the Reebvector field. A map with this property will be called basic . Next he proved that under suit-able rank conditions the harmonic map from a compact Sasakian manifold to a K¨ahler man-ifold with strongly negative curvature is CR holomorphic or CR anti-holomorphic. However,it seems that Petit [17] did not specifically discuss the relevant notions of pluriharmonicity. Keyword: CR manifolds, CR holomorphic maps, CR pluriharmonic maps, harmonic maps, pseudohar-monic maps.
1n the other hand, E. Barletta et al. in [1] introduced the so-called pseudoharmonic mapsfrom CR manifolds which are a natural generalization of harmonic maps. In his thesis [4],T.-H. Chang discussed some fundamental properties of pseudoharmonic maps.In this paper, we will establish some rigidity results for both harmoinic maps and pseu-doharmonic maps from CR manifolds by using the moving frame method. First, we find aresult about the relationship between harmonic maps and pseudoharmonic maps from CRmanifolds, which claims that these two kinds of maps are actually equivalent if the mapsare basic. By the moving frame method, we not only recapture Petit’s result about har-monic maps from compact Sasakian manifolds to Riemannian manifolds with nonpositivecurvature (Proposition 5.1), but also show that the result is still valid for pseudoharmonicmaps (Theorem 5.1).The usual Bochner-type formula for the energy density of harmonic maps was givenin [10]. In [4], T.-H. Chang derived the CR Bochner-type formula for the pseudo-energydensity of a pseudoharmonic map φ (Corollary 4.1). Unlike the Bochner formula of harmonicmaps, there is a mixed term i ( φ iα φ i ¯ α − φ i ¯ α φ iα ) appearing in the CR Bochner formula forthe pseudoharmonic map. When φ is a function, it is known that the CR Paneitz operator,which is a divergence of a third order differential operator P , is a useful tool to treat suchkind of term. One important property of the CR Paneitz operator is its nonnegativitywhen the dimension of the CR manifold ≥ P toa differential operator, still denoted by P , acting on maps from a strictly pseudoconvexCR manifolds into a Riemannian manifold, and establish similar nonnegativity under theassumptions that the domain CR manifold has dimension ≥ B and a new second fundamental form β . The later one is defined withrespect to the Tanaka-Webster connection of the domain CR manifold and the Levi-Civitaconnection of the target Riemannian manifold (see Section 2). Using B , Ianus and Pastore[13] defined two kinds of pluriharmonic notions. In [8], Dragomir and Kamishima introducedthe notion of CR pluriharmonic map by means of β . It turns out that a CR pluriharmonicmap is basic and pseudoharmonic, and thus it is harmonic too. In addition, when the targetmanifold is K¨ahler, the CR pluriharmonic maps in [8] are more compatible with the CRholomorphic maps defined in [11] in the sense that any CR holomorphic maps are automat-ically CR pluriharmonic. We also discuss the relationships between the CR pluriharmonicmaps and those defined by Ianus and Pastore. Next, using the Siu-Sampson technique, weprove that any harmonic maps or pseudoharmonic maps from compact Sasakian manifoldsto Riemannian manifolds with nonpositive Hermitian curvature or K¨ahler manifolds withstrong semi-negative curvature are CR pluriharmonic (Theorems 6.1, 6.2). If the target is2 K¨ahler manifold with strongly negative curvature and the rank of the map ≥ · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · . Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · . Preliminaries · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · . Commutative relations · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · . CR Bochner type results · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · . Basicity of harmonic and pseudoharmonic maps · · · · · · · · · · · · · · . CR pluriharmonicity of harmonic and pseudoharmonic maps ·· . Siu-Sampson type results · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · A smooth manifold M of real (2 m + 1)-dimension is said to be a CR manifold (of type(m,1)) if there exists a smooth m -dimensional complex subbundle T , M of the complexifedtangent bundle T C M = T M ⊗ C , such that T , M ∩ T , M = { } and [Γ ∞ ( T , M ) , Γ ∞ ( T , M )] ⊆ Γ ∞ ( T , M ) , where T , M = T , M . The subbundle T , M is called a CR structure on M . Equivalently,the CR structure may also be described by the real subbundle H ( M ) = Re { T , M ⊕ T , M } ,which carries a complex structure J : H ( M ) → H ( M ) given by J ( Z + ¯ Z ) = √− Z − ¯ Z )3or any Z ∈ T , M .Hereafter we assume M is orientable. Set E x = { ω ∈ T ∗ x M : Ker ( ω ) ⊇ H ( M ) x } , for any x ∈ M . Then E → M becomes an orientable real line subbundle of the cotangentbundle T ∗ M , and thus there exist globally defined nonvanishing sections θ ∈ Γ ∞ ( E ). Anysuch a section θ is called a pseudo-Hermitian structure on M . The Levi form G θ of θ isdefined by G θ ( X, Y ) = dθ ( X, J Y )for any
X, Y ∈ H ( M ). An orientable CR manifold endowed with a pseudo-Hermitianstructure is called a pseudo-Hermitian manifold. A pseudo-Hermitian manifold ( M, J, θ ) issaid to be a strictly pseudoconvex CR manifold if L θ is positive definite. Standard examplesfor strictly pseudoconvex CR manifolds are the odd-dimensional spheres and the Heisenberggroups.From now on, we always assume ( M, J, θ ) is strictly pseudoconvex. Consequently thereexists a unique nonvanishing vector field T on M , transverse to H ( M ), satisfying θ ( T ) = 1and T y dθ = 0. The vector field T is referred to as the characteristic direction or the Reebfield of ( M, J, θ ). Extending J on T M by J T = 0, we can extend L θ on T M by the sameformula as above. This allows us to define a Riemannian metric g θ , called the Webstermetric, as follows: g θ ( X, Y ) = G θ ( π H X, π H Y ) + θ ( X ) θ ( Y ) , for any X, Y ∈ T M , where π H : T M → H ( M ) is the natural projection. Then the two-formΩ defined by Ω( X, Y ) = g θ ( X, J Y ) coincides with the two-form − dθ .On a strictly pseudoconvex CR manifold, there exists a canonical connection preservingboth the CR structure and the Webster metirc. Proposition 2.1. (cf. [9, 24, 27]) Let ( M, J, θ ) be a strictly pseudoconvex CR manifoldand g θ the Webster metric of ( M, J, θ ) . Then there exists a unique linear connection ∇ on T M , called the Tanaka-Webster connection, such that:(1) the Levi distribution H ( M ) is parallel with respect to ∇ ;(2) ∇ g θ = 0 , ∇ J = 0 , ∇ θ = 0 (hence ∇ T = 0 );(3) the torsion T ∇ of ∇ satisfies T ∇ ( X, Y ) = − Ω( X, Y ) T and T ∇ ( T, J X ) = − J T ∇ ( T, X ) ,for any X, Y ∈ H ( M ) . Unlike the Levi-Civita connection, the torsion T ∇ of the Tanaka-Webster connection ∇ is always non zero. The T M -valued 1-form τ , defined by τ ( X ) = T ∇ ( T, X ), for any X ∈ T ( M ), is called the pseudo-Hermitian torsion of ∇ . Note that τ is self-adjoint andtrace-free with respect to the Webster metric g θ (cf. Chapter 1 of [9]). Definition 2.1.
A strictly pseudoconvex CR manifold is called a Sasakian manifold if itspseudo-Hermitian torsion is zero.
Choose a local orthonormal CR frame field { e = T, e , · · · , e m , J e , · · · , J e m } on M .Set T α = 1 √ e α − √− J e α ) , T ¯ α = 1 √ e α + √− J e α ) , { T α } is a local unitary frame of T , M . Let { θ, θ α , θ ¯ α } be the dual frame field of { T, T α , T ¯ α } . Clearly Proposition 2.1 implies that there exist uniquely defined complex 1-forms θ αβ ∈ Γ ∞ ( T ∗ M ) ⊗ C such that ∇ T α = θ βα ⊗ T β , ∇ T ¯ α = θ ¯ β ¯ α ⊗ T ¯ β , where θ ¯ α ¯ β = θ αβ . These are the connection 1-forms of the Tanaka-Webster connection ∇ . Set τ ( T α ) = A ¯ βα T ¯ β , and A ( T α , T β ) = g θ ( τ ( T α ) , T β ) = A αβ , then A αβ = A ¯ γα δ γβ = A ¯ βα . We denote τ α = A α ¯ β θ ¯ β , then τ = τ α ⊗ T α + τ ¯ α ⊗ T ¯ α . Write R α ¯ βλ ¯ µ = g θ ( R ( T λ , T ¯ µ ) T α , T ¯ β ) = δ γβ R βαλ ¯ µ . Lemma 2.1. (cf. [9, 27]) The structure equations for the Tanaka-Webster connection of ( M, θ, J ) in terms of local orthonormal CR coframe field { θ, θ α , θ ¯ α } are dθ = √− δ αβ θ α ∧ θ ¯ β ,dθ α = θ β ∧ θ αβ + θ ∧ τ α , θ βα + θ ¯ α ¯ β = 0 , (2.1) dθ αβ = − θ αγ ∧ θ γβ + Π αβ , where Π αβ = R αβγ ¯ δ θ γ ∧ θ ¯ δ + W αβγ θ γ ∧ θ − W αβ ¯ γ θ ¯ γ ∧ θ + √− θ β ∧ τ α − √− τ β ∧ θ α , (2.2) and W βα ¯ γ = h ¯ δβ A ¯ γ ¯ δ,α , W βαγ = h ¯ δβ A αγ, ¯ δ , τ α = h α ¯ β τ ¯ β , θ α = h α ¯ β θ ¯ β , (2.3) where R denote the curvature tensor of ∇ . From (2.1), one may derive that (cf. [27]): R α ¯ βλ ¯ µ = R λ ¯ βα ¯ µ . The pseudo-HermitianRicci tensor is given by R λ ¯ µ = R αλα ¯ µ = R ααλ ¯ µ .For a strictly pseudoconvex CR manifold ( M m +1 , J, θ ), we denote by ∇ θ the Levi-Civitaconnection of the Webster metric g θ . From Lemma 1.3 of [9], we know the relation betweenthe Tanaka-Webster connection ∇ and Levi-Civita connection ∇ θ of ( M, J, θ ): ∇ θ = ∇ + ( 12 Ω − A ) ⊗ T + τ ⊗ θ + 12 θ ⊙ J, (2.4)where A ( X, Y ) = g θ ( τ X, Y ), ( θ ⊙ J )( X, Y ) = θ ( X ) J Y + θ ( Y ) J X (cf. also [15]). By (2.4),we have ∇ θX T = τ ( X ) + 12 J X.
In particular, ∇ θT T = 0. If X, Y ∈ H ( M ), then ∇ θX Y = ∇ X Y + [ 12 Ω( X, Y ) − A ( X, Y )] T. (2.5) Lemma 2.2.
For any local orthonormal CR frame field { e A } mA =0 , we have m X A =0 ∇ θe A e A = m X A =0 ∇ e A e A . (2.6) In particular, we get m X A =1 ∇ θe A e A = m X A =1 ∇ e A e A . (2.7)5 roof. By (2.4), we have n X A =0 ∇ θe A e A − n X A =0 ∇ e A e A = − trace ( τ ) T = 0 . Since ∇ θT T = ∇ T T = 0, (2.7) is valid.As a result of Lemma 2.2, we have Lemma 2.3.
Let ( M, J, θ ) be a strictly pseudoconvex CR manifold and let X be any vectorfield on M . Then divX = n X A =0 g θ ( ∇ e A X, e A ) . where ∇ is the Tanaka-Webster connection of M and { e A } mA =0 is a local orthonormal CRframe field on M . In particular, if X ∈ H ( M ) , then divX = n X A =1 g θ ( ∇ e A X, e A ) . Let (
M, J, θ ) be a strictly pseudoconvex CR manifold with the Tanaka-Webster connec-tion ∇ and let ( N, h ) be a Riemannian manifold with Levi-Civita connection ∇ h . For asmooth map φ : M → N , there are two induced connections ∇ θ ⊗ φ − ∇ h and ∇ ⊗ φ − ∇ h on T ∗ M ⊗ φ − T N . Using these two connections, one may define the usual second fundamentalform B and a new second fundamental form β (cf. [17]) for the map φ as follows: B ( X, Y ) = ∇ hY ( dφ ( X )) − dφ ( ∇ θY X ) (2.8)and β ( X, Y ) = ∇ hY ( dφ ( X )) − dφ ( ∇ Y X ) , (2.9)where φ − ∇ h is written as ∇ h for simplicity. Due to Lemma 2.2, we have trace g θ B = trace g θ β. (2.10)Recall that a map φ is called harmonic if τ θ ( φ ) := trace g θ B = 0 (cf. [10]). As a result of(2.10), the harmonicity of φ can also be defined by β . Note that the most advantage ofusing ∇ in (2.9) is that the Tanaka-Webster connection preserves the CR structure; a littledisadvantage of using ∇ is that β is no longer symmetric. However, we will see that thenon-symmetry of β may also lead to some unexpected geometric consequences.For any bilinear form C on T M , we denote by π H C the restriction of C to H ( M ) ⊗ H ( M ). Definition 2.2.
A map φ : ( M m +1 , J, θ ) → ( N, h ) from a strictly pseudoconvex CR man-ifold to a Riemannian manifold is called a pseudoharmonic map if it is a critical point ofthe following pseudo-energy functional E H ( φ ) = Z M e H ( φ )Ψ (2.11) where e H ( φ ) = trace G θ ( π H φ ∗ h ) is the pseudo-energy density of φ and Ψ = θ ∧ ( dθ ) m isthe volume form of g θ . roposition 2.2. (cf. [1, 9]) Let φ : ( M m +1 , J, θ ) → ( N, h ) be a smooth map from astrictly pseudoconvex CR manifold to a Riemannian manifold. Let τ ( φ ) be pseudo-tensorfield of φ defined by τ ( φ ) = trace G θ ( π H β ) . (2.12) Then φ is pseudoharmonic if and only if τ ( φ ) = 0 . From Lemma 2.2, it is easy to see that τ ( φ ) = trace G θ ( π H B ) . (2.13) Definition 2.3.
A smooth map φ : ( M m +1 , J, θ ) → ( N, h ) is called basic if dφ ( T ) = 0 . Proposition 2.3.
Let φ : ( M, J, θ ) → ( N, h ) be a smooth map. Assume that ∇ hT ( dφ ( T )) =0 , that is, dφ ( T ) is parallel in the direction T with respect to the pull-back connection φ − ∇ h .Then τ θ ( φ ) = τ ( φ ) ; and thus φ is harmonic if and onlu if φ is pseudoharmonic.Proof. Choose a local orthonormal CR frame field { e A } nA =0 = { T, e , e · · · , e n } . UsingLemma 2.2 and the assumption, we compute τ θ ( φ ) = n X A =1 [ ∇ he A ( dφ ( e A )) − dφ ( ∇ θe A e A )] + ∇ hT ( dφ ( T ))= n X A =1 [ ∇ he A ( dφ ( e A )) − dφ ( ∇ e A e A )]= τ ( φ ) . (2.14) Corollary 2.1.
Let φ : ( M m +1 , J, θ ) → ( N, h ) be a baisc map. Then φ is harmonic if andonly if φ is pseudoharmonic. Definition 2.4.
Let φ : ( M, J, θ ) → ( N, h ) be a smooth map from a strictly pseudoconvexCR manifold into a Riemannian manifold. We say that(i) ([13]) φ is J-pluriharmonic, if B ( X, Y ) + B ( J X, J Y ) = 0 , for any
X, Y ∈ T M (ii) ([13]) φ is H-pluriharmonic, if B ( Z, W ) + B ( J Z, J W ) = 0 , for any
Z, W ∈ H ( M ) ;(iii) φ is B-pluriharmonic, if B ( T, T ) = 0 and B ( Z, W ) + B ( J Z, J W ) = 0 , for any
Z, W ∈ H ( M ) ;(iv) ([8]) φ is CR pluriharmonic, if β ( Z, W ) + β ( J Z, J W ) = 0 , for any
Z, W ∈ H ( M ) ;(v) ([11]) When ( N, h ) is a K¨ahler manifold with complex structure J ′ , φ is called a CRholomorphic (resp. CR anti-holomorphic) map, if dφ ◦ J = J ′ ◦ dφ, ( resp. dφ ◦ J = − J ′ ◦ dφ ) . (2.15) Remark 2.1. (1) The concepts of J-pluriharmonic map and H-pluriharmonic map wereintroduced by Ianus and Pastore in [13] where J and H ( M ) are denoted by ϕ and D respec-tively. And they proved that the J-pluriharmonic maps are harmonic.(2) Dragomir and Kamishima in [8] introduced the notion of CR pluriharmonic mapsunder the name of ¯ ∂ -pluriharmonic map, and then they proved that every CR pluriharmonicmap is a pseudoharmonic map.
3) In [11] the authors introduced the notion of CR holomorphic map under the nameof the ( J, J ′ ) -holomorphic map. They proved that the CR holomorphic map is harmonic. If ( M, g, J ) is a K¨ahler manifold, ( N, h ) is a Riemannian manifold and the map φ : M → N satisfies B ( X, Y ) + B ( J X, J Y ) = 0 , for any X, Y ∈ T M , then the map φ is called a pluriharmonic map (cf. [6, 26]). Obviously, J-pluriharmonicity implies B-pluriharmonicity, and B-pluriharmonicity im-plies H-pluriharmonicity. Both J-pluriharmonic maps and B-pluriharmonic maps are har-monic. By (2.12) and (2.13), both the CR pluriharmonic map and the H-pluriharmonicmap are pseudoharmonic.
Proposition 2.4. (i) (cf. [8]) If φ : ( M, J, θ ) → ( N, h ) is CR pluriharmonic, then φ is abasic and pseudoharmonic map. Moreover, φ is B-pluriharmonic too.(ii) If φ : ( M, J, θ ) → ( N, h ) is baisc and H-pluriharmonic, then φ is CR pluriharmonic.Proof. (i) For any Z = X − √− J X, W = Y − √− J Y ∈ T , M , we have β ( Z, W ) = β ( X, Y ) + β ( J X, J Y ) + √− β ( X, J Y ) − β ( J X, Y )] , (2.16)thus we get that φ is CR pluriharmonic if and only if ( π H β ) (1 , = 0. Thus the CRpluriharmonic map is pseudoharmonic.On the other hand, we have0 = β ( Z, W ) − β ( W , Z )= dφ ( T ∇ ( Z, W ))= − Ω( Z, W ) dφ ( T )= √− g θ ( Z, W ) dφ ( T ) . (2.17)If we take Z = W = 0, then g θ ( Z, W ) = 0, thus we have dφ ( T ) = 0.For any X, Y ∈ H ( M ), by (2.5) and A ( J Y, J X ) = A ( Y, X ), we have B ( X, Y ) + B ( J X, J Y ) = β ( X, Y ) + β ( J X, J Y ) − Ω( Y, X ) dφ ( T ) . (2.18)Thus if φ is baisc, then the CR pluriharmonic map φ is B-pluriharmonic.(ii) This can be proved by (2.18). Proposition 2.5.
Suppose φ : ( M, J, θ ) → ( N, h, J ′ ) is a CR ± holomorphic map from astrictly pseudoconvex CR manifold M into a K¨ahler manifold N . Then φ is CR plurihar-monic.Proof. Suppose φ is CR holomorphic map. For any X, Y ∈ H ( M ), we have β ( J X, Y ) = ∇ hY ( φ ( J X )) − dφ ( ∇ Y J X )= ∇ hY ( J ′ dφ ( X )) − dφ ( J ( ∇ Y X ))= J ′ ∇ hY ( dφ ( X )) − J ′ dφ ( ∇ Y X )= J ′ β ( X, Y ) . (2.19)8ince J ′ dφ ( T ) = dφ ( J T ) = 0, we get that φ is baisc. Because of β ( X, Y ) − β ( Y, X ) = − Ω( X, Y ) dφ ( T ), we have that β is symmetric on H ( M ) ⊗ H ( M ). Thus we derive β ( J X, J Y ) = J ′ β ( X, J Y ) = J ′ β ( J Y, X ) = − β ( Y, X ) = − β ( X, Y ) . Therefore, the map φ is CR pluriharmonic. If φ is CR anti-holomorphic map, the conclusioncan be proved in a similar way. Let φ : ( M m +1 , J, θ ) → ( N n , h ) be a smooth map from a strictly pseudoconvex CR man-ifold into a Riemannian manifold. Choose a local orthonormal CR coframe field { θ, θ α , θ ¯ α } on M and a local orthonormal coframe field { ω i } on N . Throughout this paper we willemploy the index conventions A, B, C = 0 , , · · · , m, ¯1 , · · · , ¯ m,α, β, γ = 1 , · · · , m,i, j, k = 1 , · · · , n, and use the summation convention on repeating indices. The structure equations for theRiemannian connection of ( N, h ) in terms of local orthonormal frame { ω i } are dω i = − ω ij ∧ ω j , ω ij + ω ji = 0 ,dω ij = − ω ik ∧ ω kj + Ω ij , (3.1)where Ω ij = fi R ijkl ω k ∧ ω l are the components of the curvature form of ∇ h .Under the map φ : M → N , we have φ ∗ ω i = φ iα θ α + φ i ¯ α θ ¯ α + φ i θ. (3.2)Hereafter we will drop φ ∗ in such formulas when their meaning are clear from context. Bytaking the exterior derivative of (3.2) and making use of the structure equations (2.1)-(2.3)and (3.1), we get Dφ iB ∧ θ B + √− φ i θ α ∧ θ ¯ α − φ iα A ¯ α ¯ β θ ¯ β ∧ θ − φ i ¯ α A αβ θ β ∧ θ = 0 , (3.3)where Dφ iα = dφ iα − φ iβ θ βα + φ jα ω ij = φ iαB θ B , (3.4) Dφ i ¯ α = dφ i ¯ α − φ i ¯ β θ ¯ β ¯ α + φ j ¯ α ω ij = φ i ¯ αB θ B , (3.5) Dφ i = dφ i + φ j ω ij = φ i B θ B . (3.6)From (3.3) it follows that φ iαβ = φ iβα , φ iα ¯ β − φ i ¯ βα = √− δ αβ φ i , φ i α − φ iα = φ i ¯ β A βα . (3.7)9hen the map φ is harmonic if and only if φ iα ¯ α + φ i ¯ αα + φ i = 0 , and φ is pseudoharmonic if and only if φ iα ¯ α + φ i ¯ αα = 0 . Differentiating the equation (3.4) and using the structure equations in M and N , wehave Dφ iαB ∧ θ B + √− φ iα θ β ∧ θ ¯ β − φ iαβ A ¯ β ¯ γ θ ¯ γ ∧ θ − φ iα ¯ β A βγ θ γ ∧ θ = − φ iβ Π βα + φ jα Ω ij , (3.8)where Dφ iαβ = dφ iαβ − φ iαγ θ γβ − φ iγβ θ γα + φ jαβ ω ij = φ iαβB θ B ,Dφ iα ¯ β = dφ iα ¯ β − φ iα ¯ γ θ ¯ γ ¯ β − φ iγ ¯ β θ γα + φ jα ¯ β ω ij = φ iα ¯ βB θ B ,Dφ iα = dφ iα − φ iγ θ γα + φ jα ω ij = φ iα B θ B . From (3.8), we get the following commutative relations φ iαβγ = φ iαγβ − φ jα φ kβ φ lγ ‘ R ijkl + √− φ iβ A αγ − √− φ iγ A αγ , (3.9) φ iα ¯ β ¯ γ = φ iα ¯ γ ¯ β − φ jα φ k ¯ β φ l ¯ γ ‘ R ijkl + √− δ αβ φ iλ A ¯ λ ¯ γ − √− δ αγ φ iλ A ¯ λ ¯ β , (3.10) φ iαβ ¯ γ = φ iα ¯ γβ − φ jα φ kβ φ l ¯ γ ‘ R ijkl + φ iλ R λαβ ¯ γ + √− δ βγ φ iα , (3.11) φ iαβ = φ iα β − φ jα φ kβ φ l ‘ R ijkl + φ iγ A αβ,γ − φ iα ¯ γ A γβ , (3.12) φ iα ¯ β = φ iα β − φ jα φ k ¯ β φ l ‘ R ijkl − φ iγ A ¯ β ¯ γ,α − φ iαγ A ¯ γ ¯ β , (3.13)where ‘ R ijkl = fi R ijkl ◦ φ .Since the formula (3.5) is the complex conjugate of (3.4), then, after taking the exteriorderivative of (3.5) and using the structure equations, we find that the complex conjugate offormulas (3.9)-(3.13) are valid too.Similarly the exterior derivative of (3.6) yields that Dφ i B ∧ θ B + √− φ i θ α ∧ θ ¯ α − φ iα A ¯ α ¯ β θ ¯ β ∧ θ − φ i ¯ α A αβ θ β ∧ θ = φ j Ω ij , (3.14)where Dφ i α = dφ i α − φ i β θ βα + φ j α ω ij = φ i αB θ B ,Dφ i α = dφ i α − φ i β θ ¯ β ¯ α + φ j α ω ij = φ i αB θ B ,Dφ i = dφ i + φ j ω ij = φ i B θ B . We get from (3.14) the commutative relations φ i αβ = φ i βα − φ j φ kα φ lβ ‘ R ijkl , (3.15) φ i α ¯ β = φ i βα − φ j φ kα φ l ¯ β ‘ R ijkl + √− δ αβ φ i , (3.16) φ i α = φ i α − φ j φ k φ lα ‘ R ijkl + φ i β A βα . (3.17)10rom (3.7), we can derive: φ iα ¯ βγ = φ i ¯ βαγ + √− δ αβ φ i γ , (3.18) φ iα ¯ β ¯ γ = φ i ¯ βα ¯ γ + √− δ αβ φ i γ , (3.19) φ i αβ = φ iα β + φ i ¯ γβ A γα + φ i ¯ γ A γα,β , (3.20) φ i α ¯ β = φ iα β + φ i ¯ γ ¯ β A γα + φ i ¯ γ A γα, ¯ β . (3.21)If ( N, h ) is a K¨ahler manifold, we choose a local orthonormal coframe field { ω i , ω ¯ i } on N . The structure equations for the Riemannian connection of ( N, h ) in terms of localorthonormal frame { ω i , ω ¯ i } are dω i = − ω ij ∧ ω j , ω ij + ω ¯ j ¯ i = 0 ,dω ij = − ω ik ∧ ω kj + Ω ij , (3.22)where Ω ij = fi R ijkl ω k ∧ ω ¯ l . Similar to the above discussions, we may obtain the followingcommutative formula: φ iα ¯ β ¯ γ = φ iα ¯ γ ¯ β − φ jα φ k ¯ β φ ¯ l ¯ γ ‘ R ijkl + φ jα φ k ¯ γ φ ¯ l ¯ β ‘ R ijkl + √− δ αβ φ iλ A ¯ λ ¯ γ − √− δ αγ φ iλ A ¯ λ ¯ β . (3.23) Let ( M m +1 , J, θ ) be a compact strictly pseudoconvex CR manifold. In [12, 14] theauthors introduced the following differential operator acting on functions P f = X ( f ¯ ααβ + √− mA βα f ¯ α ) θ β = ( P β f ) θ β , which charecterizes CR pluriharmonic functions on M . In [3] S.-C. Chang and H.-L.Chiudiscussed the CR Paneitz operator P f = 4[ δ b ( P f ) + ¯ δ b ( ¯ P f )] , where δ b is the divergence operator that take (1 , m ≥
2, the corresponding CR Paneitz operator is always nonnegative, that is Z M P f · f Ψ ≥ , where Ψ is the volume form of g θ .Now we want to generalize the operator P to an operator, still denoted by P , acting onmaps from strictly pseudoconvex CR manifolds into Riemannian manifolds. We will estab-lish similar nonnegative property for the generalized operator P under suitable condition.Suppose φ : ( M m +1 , θ, J ) → ( N n , h ) is a smooth map from a strictly pseudoconvex CRmanifold M into a Riemannian manifold N . We choose a local orthonormal CR coframefield { θ, θ α , θ ¯ α } on M , a local orthonormal frame field { E i } on N . We still use the notaionsof the last section. Define P φ = ( P jβ φ ) θ β ⊗ E j , P jβ φ = φ j ¯ ααβ + √− mA βα φ j ¯ α .Let θ W = φ iα φ i ¯ α ¯ β θ ¯ β + φ i ¯ α φ iαβ θ β + φ iα φ i ¯ αβ θ β + φ i ¯ α φ iα ¯ β θ ¯ β . (4.1)Evidently the 1-form θ W , which is a well-defined on M , is the 1-form correspondingto the horizontal gradient ∇ H ( e H ( φ )) of e H ( φ ), where ∇ H ( e H ( φ )) = Π H ∇ ( e H ( φ )) and g θ ( ∇ e H ( φ ) , X ) = X ( e H ( φ )) for any X ∈ χ ( M ). Lemma 4.1.
Set fl R ijkl = g ip fi R pjkl = δ ip fi R pjkl = fi R ijkl . Then divθ W = 2( | φ iαβ | + | φ iα ¯ β | ) + hh d b φ, ∇ b τ ( φ ) ii + 2 φ iα φ i ¯ β Ric ¯ αβ −√− m ( φ iα φ iβ A ¯ α ¯ β − φ i ¯ α φ i ¯ β A αβ ) − √− φ iα φ i ¯ α − φ i ¯ α φ iα ) − φ i ¯ α φ jβ φ kα φ l ¯ β ’ R ijkl + φ iα φ jβ φ k ¯ α φ l ¯ β ’ R ijkl ) , (4.2) where ∇ b τ ( φ ) = ( φ iα ¯ αβ + φ i ¯ ααβ ) θ β ⊗ E i + ( φ iα ¯ α ¯ β + φ i ¯ αα ¯ β ) θ ¯ β ⊗ E i , and hh· , ·ii is the metric in T ∗ M ⊗ φ − T N induced by g θ and h .Proof. Using Lemma 2.3 and the commutative relations in Section 3, we compute divθ W = ( φ iα φ i ¯ α ¯ β ) , β +( φ i ¯ α φ iαβ ) , ¯ β +( φ iα φ i ¯ αβ ) , ¯ β +( φ i ¯ α φ iα ¯ β ) , β = φ iαβ φ i ¯ α ¯ β + φ iα φ i ¯ α ¯ ββ + φ i ¯ α ¯ β φ iαβ + φ i ¯ α φ iαβ ¯ β + φ iα ¯ β φ i ¯ αβ + φ iα φ i ¯ αβ ¯ β + φ i ¯ αβ φ iα ¯ β + φ i ¯ α φ iα ¯ ββ = 2( | φ iαβ | + | φ iα ¯ β | ) + hh d b φ, ∇ b τ ( φ ) ii + 2 φ iα φ i ¯ β Ric ¯ αβ − √− m ( φ iα φ iβ A ¯ α ¯ β − φ i ¯ α φ i ¯ β A αβ ) − √− φ iα φ i ¯ α − φ i ¯ α φ iα ) − φ i ¯ α φ jβ φ kα φ l ¯ β ’ R ijkl + φ iα φ jβ φ k ¯ α φ l ¯ β ’ R ijkl ) . From (3.7), (3.11) and (3.19), we get immediately the following Lemmas 4.2, 4.3 and4.4.
Lemma 4.2. √− φ iα φ i ¯ α − φ i ¯ α φ iα ) = 2 m hh P φ + P φ, d b φ ii − m hh d b φ, ∇ b τ ( φ ) ii + √− φ iα φ iβ A ¯ α ¯ β − φ i ¯ α φ i ¯ β A αβ ) . (4.3)Thus we have Corollary 4.1. divθ W = 2( | φ iαβ | + | φ iα ¯ β | ) + (1 + 2 m ) hh d b φ, ∇ b τ ( φ ) ii + 2 φ iα φ i ¯ β Ric ¯ αβ −√− m + 2)( φ iα φ iβ A ¯ α ¯ β − φ i ¯ α φ i ¯ β A αβ ) − m hh P φ + P φ, ∇ b φ ii− φ i ¯ α φ jβ φ kα φ l ¯ β ’ R ijkl + φ iα φ jβ φ k ¯ α φ l ¯ β ’ R ijkl ) , (4.4) Remark 4.1.
Since divθ W = ∆ b ( e H ( φ )) , the formula (4.2) and (4.4) are both called theCR Bochner formulae. √− Z M ( φ iα φ i ¯ α − φ i ¯ α φ iα )Ψ = 2 m Z M hh P φ + P φ, d b φ ii Ψ + 1 m Z M | τ ( φ ) | Ψ+ √− Z M ( φ iα φ iβ A ¯ α ¯ β − φ i ¯ α φ i ¯ β A αβ )Ψ . (4.5) Lemma 4.3. √− Z M ( φ iα φ i ¯ α − φ i ¯ α φ iα )Ψ = m Z M ( φ i ) Ψ − √− Z M ( φ iα φ iβ A ¯ α ¯ β − φ i ¯ α φ i ¯ β A αβ )Ψ . (4.6) Lemma 4.4. Z M φ iα φ i ¯ β Ric ¯ αβ Ψ = − Z M ( | φ iαβ | − | φ iα ¯ β | )Ψ + √− m Z M ( φ iα φ i ¯ α − φ i ¯ α φ iα )Ψ+2 Z M φ iα φ j ¯ α φ k ¯ β φ lβ ’ R ijkl Ψ . (4.7)Integrating (4.2) on M and substituting (4.7) into it, we have0 = 4 Z M | φ iα ¯ β | Ψ − Z M | τ ( φ ) | Ψ + √− m − Z M ( φ iα φ i ¯ α − φ i ¯ α φ iα )Ψ −√− m Z M ( φ iα φ iβ A ¯ α ¯ β − φ i ¯ α φ i ¯ β A αβ )Ψ − Z M ( φ i ¯ α φ jβ φ kα φ l ¯ β ’ R ijkl + φ iα φ jβ φ k ¯ α φ l ¯ β ’ R ijkl − φ iα φ j ¯ α φ k ¯ β φ lβ ’ R ijkl )Ψ . By the Bianchi identity, we find − φ iα φ j ¯ α φ k ¯ β φ lβ ’ R ijkl = φ iα φ j ¯ α φ k ¯ β φ lβ ( ’ R iklj + ’ R iljk )= φ iα φ j ¯ β φ kβ φ l ¯ α ’ R ijkl + φ iα φ jβ φ k ¯ α φ l ¯ β ’ R ijkl = − φ i ¯ α φ jβ φ kα φ l ¯ β ’ R ijkl + φ iα φ jβ φ k ¯ α φ l ¯ β ’ R ijkl . Hence 0 = 4 Z M | φ iα ¯ β | Ψ − Z M | τ ( φ ) | Ψ + √− m − Z M ( φ iα φ i ¯ α − φ i ¯ α φ iα )Ψ −√− m Z M ( φ iα φ iβ A ¯ α ¯ β − φ i ¯ α φ i ¯ β A αβ )Ψ − Z M φ iα φ jβ φ k ¯ α φ l ¯ β ’ R ijkl Ψ . (4.8)Calculating (4.5) × ( m − − (4.6) and substituting the result into the above formula, wehave0 = 4 Z M | φ iα ¯ β | Ψ − m Z M | τ ( φ ) | Ψ − m Z M ( φ i ) Ψ + 2( m − m Z M hh P φ + P φ, d b φ ii Ψ − Z M φ iα φ jβ φ k ¯ α φ l ¯ β ’ R ijkl Ψ . (4.9)Since | φ iα ¯ β | ≥ m | X φ iα ¯ α | = 14 m | τ ( φ ) | + m φ i ) , we conclude − Z M hh P φ + P φ, d b φ ii Ψ ≥ − mm − Z M φ iα φ jβ φ k ¯ α φ l ¯ β ’ R ijkl Ψ . (4.10)13 efinition 4.1. (cf. [21]) A Riemannian manifold ( N n , h ) is said to have nonpositiveHermitian curvature if R ijkl u i v j ¯ u k ¯ v l Ψ ≤ , (4.11) for any complex vectors u and v . From (4.10), we have
Theorem 4.1.
Let ( M m +1 , J, θ ) be a compact strictly pseudoconvex CR manifold with m ≥ and ( N, h ) be a Riemannian manifold with nonpositive Hermitian curvature. Suppose φ : M → N is a smooth map, then − Z M h P φ + P φ, d b φ i Ψ ≥ . Let’s denote
Ric ( X, Y ) = R α ¯ β X α Y ¯ β ,T or ( X, Y ) = √− A ¯ α ¯ β X ¯ α Y ¯ β − A αβ X α Y β ) , where X = X α T α , Y = Y β T β and R α ¯ β = R γγα ¯ β is the pseudo-Hermitian Ricci curvature of M . We denote ( ∇ b φ i ) C = φ iα T α . Theorem 4.2.
Let ( M m +1 , J, θ ) be a compact strictly pseudoconvex CR manifold with m ≥ and ( N, h ) be a Riemannian manifold with nonpositive Hermitian curvature. Let φ : M → N be a pseudoharmonic map. Suppose that Ä Ric − ( m + 2) T or ä ( Z, Z ) ≥ , (4.12) for any Z ∈ Γ ∞ ( T , M ) , then(i) φ is horizontal totally geodesic, that is φ iαβ = φ iα ¯ β = 0 . In particular, φ is baisc;(ii)If Ä Ric − ( m + 2) T or ä ( Z, Z ) > at one point in M , then φ is constant.Proof. (i) By (4.4), we have0 = 2 Z M ( | φ iαβ | + | φ iα ¯ β | )Ψ − (1 + 2 m ) Z M | τ ( φ ) | Ψ − m Z M h P φ + P φ, d b φ i Ψ+ Z M (2 Ric − ( m + 2) T or )(( ∇ b φ i ) C , ∇ b φ i ) C )Ψ − Z M ( φ i ¯ α φ jβ φ kα φ l ¯ β ’ R ijkl + φ iα φ jβ φ k ¯ α φ l ¯ β ’ R ijkl )Ψ . By Theorem 4.1, the CR Paneitz operator from M into N is nonnegative. Because of thecurvature condition of N , the last term of the above formula is nonnegative. Since (4.11)and φ is pseudoharmonic, we get0 ≥ Z M ( | φ iαβ | + | φ iα ¯ β | )Ψ . Hence φ iαβ = φ iα ¯ β = 0. From φ iα ¯ β = 0, we see that φ is CR pluriharmonic, so φ is baisc.(ii) Since φ is baisc and pseudoharmonic, by Proposition 2.3, we have that φ is harmonic.By the curvature condition of M , we have ( ∇ b φ i ) C = 0 in some neigberhood U of that point.Thus we get φ is constant in U . It follows from the unique continuation theorem (cf. [20])that φ is constant on M . 14 emark 4.2. If the manifold M in Theorem 4.2 is Sasakian and Ric ( Z, Z ) ≥ , we have β ≡ . Suppose φ : ( M m +1 , J, θ ) → ( N, h ) is a smoooth map from a strictly pseudoconvexCR manifold into a Riemannian manifold. We choose the orthonormal CR coframe field { θ, θ α , θ ¯ α } on M and the orthonormal coframe field { ω i } on N respectively. We still usethe notaions in Section 3. Set θ W = ( φ i φ i α θ α + φ i φ i α θ ¯ α ) + φ i φ i θθ W = ( φ i φ i α θ α + φ i φ i α θ α )Clearly θ W , θ W are well-defined global 1-forms on M . In fact, θ W is the 1-form correspond-ing to the vector field ∇| dφ ( T ) | and θ W is the 1-form corresponding to the horizontalgradient ∇ H | dφ ( T ) | = Π H ∇| dφ ( T ) | .By the commutative relations in section 3, we have Lemma 5.1. divθ W = 2 | φ i α | + | φ i | + φ i ( φ iα ¯ α + φ i ¯ αα + φ i ) − φ i φ jα φ k φ l ¯ α ’ R ijkl +2 φ i φ iβ A ¯ β ¯ α,α + 2 φ i φ i ¯ β A βα, ¯ α + 2 φ i φ iαβ A ¯ β ¯ α + 2 φ i φ i ¯ α ¯ β A βα ; (5.1) divθ W = 2 | φ i α | + φ i ( φ iα ¯ α + φ i ¯ αα ) − φ i φ jα φ k φ l ¯ α ’ R ijkl +2 φ i φ iβ A ¯ β ¯ α,α + 2 φ i φ i ¯ β A βα, ¯ α + 2 φ i φ iαβ A ¯ β ¯ α + 2 φ i φ i ¯ α ¯ β A βα . (5.2) Remark 5.1.
In fact, divθ W = ∆ | dφ ( T ) | , and divθ W = ∆ b | dφ ( T ) | . Definition 5.1.
Let φ : M → N be a smooth map from a strictly pseudoconvex CR man-ifold M into a Riemannian manifold N . The second fundamental form β is called split if β ( T, X ) = 0 for any X ∈ H ( M ) . Remark 5.2.
According to (3.7), the condition β ( T, X ) = 0 for X ∈ H ( M ) is not equiva-lent to β ( X, T ) = 0 for X ∈ H ( M ) in general. From Proposition 2.1 and (2.9), it is easyto see that if φ is baisc, then the second fundamental form β is split. The next result showsthat if the domain CR manifold is compact, the converse is also true. Lemma 5.2.
Let φ : M → N be a smooth map from a compact strictly pseudoconvex CRmanifold M into a Riemannian manifold N . If the second fundamental form β is split, then φ is basic.Proof. By the integration by parts and the commutative formulae (3.7), we have0 = √− Z M ( φ iα φ i α − φ i ¯ α φ i α )Ψ = −√− Z M ( φ iα ¯ α φ i − φ i ¯ αα φ i )Ψ = m Z M | φ i | Ψ . Thus we have φ i = 0, i.e., dφ ( T ) = 0.First, we prove the following result of Petit by the moving frame method.15 roposition 5.1. (cf. [17]) Let ( M m +1 , J, θ ) be a compact Sasakian manifold and ( N, h ) be a Riemannian manifold with nonpositive curvature. Suppose φ : M → N is a harmonicmap. Then φ is basic.Proof. Since φ is harmonic, we have Dτ θ ( φ ) = 0. Consequently, φ iα ¯ α + φ i ¯ αα + φ i = 0.The Sasakian condition for M means that A αβ = 0, for any α, β , then (5.1) becomes divθ W = 2 | φ i α | + | φ i | − φ i φ jα φ k φ l ¯ α ’ R ijkl . Since the sectional curvature of N is nonpositive, we take T α = √ ( e α − iJ e α ) and T ¯ α = √ ( e α + iJ e α ) and compute the following curvature term to find φ i φ jα φ k φ l ¯ α ’ R ijkl = h ( “ R ( dφ ( T ) , dφ ( T ¯ α )) dφ ( T α ) , dφ ( T ))= 12 h ( “ R ( dφ ( T ) , dφ ( e α + iJ e α )) dφ ( e α − iJ e α ) , dφ ( T ))= 12 [ h ( “ R ( dφ ( T ) , dφ ( e α )) dφ ( e α ) , dφ ( T )) + h ( “ R ( dφ ( T ) , dφ ( J e α )) dφ ( J e α ) , dφ ( T ))] ≤ . Therefore divθ W ≥ | φ i α | + | φ i | . (5.3)The divergence theorem yields φ i = φ i α = φ i α = 0 . The fact that φ is basic can be easily obtained by Lemma 5.2.The next result shows that Petit type result is also true for pseudoharmonic maps. Theorem 5.1.
Let ( M m +1 , J, θ ) be a compact Sasakian manifold and ( N, h ) be a Rieman-nian manifold with nonpositive curvature. Suppose φ : M → N is a pseudoharmonic map.Then φ is basic and harmonic.Proof. Since φ is pseudoharmonic, we get φ iα ¯ α + φ i ¯ αα = 0 . By (5.2), we have divθ W ≥ | φ i α | (5.4)Thus φ i α = φ i α = 0. By Lemma 5.2 again, we get dφ ( T ) = 0. Remark 5.3.
From Proposition 5.1 and Theorem 5.1, we see that if M is a compactSasakian manifold and N is a Riemannian manifold with nonpositive curvature, then φ : M → N is harmonic if and only if it is pseudoharmonic. Now we will use a technique in [18] to treat harmonic maps or pseudoharmonic mapsfrom complete noncompact CR manifolds. 16 roposition 5.2.
Let ( M, J, θ ) be a complete noncompact Sasakian manifold of dimension m + 1 and ( N, h ) be a Riemannian manifold with nonpositive curvature. Suppose φ : M → N is either a harmonic map or a pseudoharmonic map. If φ satisfies ( Z ∂B r | dφ ( T ) | dS ) − / ∈ L (+ ∞ ) , (5.5) where dS is the area volume of ∂B r , then φ has split second fundamental form β .Proof. We consider only the case φ is a harmonic map, because the other case is analogous.By the divengence theorem, (5.3) gives Z ∂B r θ W ( ∂∂r ) dS ≥ Z B r (2 | φ i α | + | φ i | )Ψ . (5.6)Recalling the definition of θ W we have Z ∂B r θ W ( ∂∂r ) dS ≤ { Z ∂B r | φ i | dS } { Z ∂B r [2 | φ i α | + | φ i | ] dS } . (5.7)Let ζ ( r ) = Z B r (2 | φ i α | + | φ i | )Ψ . Then by the co-area formula, we get ζ ′ ( r ) = Z ∂B r (2 | φ i α | + | φ i | ) dS. Putting together (5.6) and (5.7) and squaring we finally get ζ ( r ) ≤ ( Z ∂B r | φ i | dS ) ζ ′ ( r ) . (5.8)Next, we reason by contradiction and we suppose φ i α = 0. It follows that there exists a R > ζ ( r ) >
0, for every r ≥ R . Fix such an r . From (5.8) wethen derive ζ ( R ) − − ζ ( r ) − ≥ Z rR dt R ∂B t | φ i | , and letting r → + ∞ we contradict (5.5). Corollary 5.1.
Let ( M, J, θ ) be a complete noncompact Sasakian manifold of dimension m + 1 and ( N, h ) be a Riemannian manifold with nonpositive curvature. Suppose φ : M → N is either a harmonic map or a pseudoharmonic map. If φ satisfies Z B r | dφ ( T ) | Ψ ≤ Cr , (5.9) then φ has split second fundamental form β ( φ ) .Proof. Set h ( r ) = Z B r | dφ ( T ) | Ψ . h ′ ( r ) = Z ∂B r | dφ ( T ) | dS. From Proposition 3.1 of [19], we know that rh ( r ) / ∈ L (+ ∞ ) implies h ′ ( r ) / ∈ L (+ ∞ ) . Suppose that φ satisfies (5.9), this implies rh ( r ) / ∈ L (+ ∞ ) . Thus we deduce h ′ ( r ) / ∈ L (+ ∞ ), that is, φ satisfies (5.5). Hence we prove the corollary. Proposition 5.3.
Let φ : ( M m +1 , J, θ ) → ( N, h ) be a smooth map from a complete non-compact strictly pseudoconvex CR manifold M into a Riemannian manifold N . If the secondfundamental form β is split and ( Z ∂B r e H ( φ ) dS ) − / ∈ L (+ ∞ ) , (5.10) then φ is basic.Proof. Since φ has split second fundamental form β , we have m Z B r | φ i | Ψ = −√− Z B r div ( φ i φ iα θ α − φ i φ i ¯ α θ ¯ α )Ψ ≤ { Z ∂B r | φ i | dS } / { Z ∂B r | φ iα | dS } / . Set η ( r ) = R B r | φ i | Ψ. Then we have m η ( r ) ≤ ( Z B r e H ( φ )Ψ) η ′ ( r ) . If φ is not basic, then for r > R , η ( R ) − − η ( r ) − ≥ Z rR dt R ∂B t e H ( φ ) dS , where R is large enough such that η ( R ) >
0, and letting r → + ∞ we contradict (5.10). Theorem 5.2.
Let ( M m +1 , J, θ ) be a complete noncompact Sasakian manifold and ( N, h ) be a Riemannian manifold with nonpositive curvature. Suppose φ : M → N is either aharmonic map or a pseudoharmonic map. If φ satisfies ( Z ∂B r e ( φ ) dS ) − / ∈ L (+ ∞ ) , (5.11) where e ( φ ) = trace g θ ( φ ∗ h ) is the energy density of φ , then φ is a basic map. roof. Since e ( φ ) = | dφ ( T ) | + e H ( φ ), the condition (5.11) implies both (5.5) and (5.10).It follows from Proposition 5.2 and 5.3 that φ is basic. Corollary 5.2.
Let ( M, J, θ ) be a complete noncompact Sasakian manifold of dimension m + 1 and ( N, h ) be a Riemannian manifold with nonpositive curvature. Suppose φ : M → N is either a harmonic map or a pseudoharmonic map. If φ satisfies Z B r e ( φ )Ψ ≤ Cr , (5.12) then φ is basic. In this section, we give some conditions to ensure the CR pluriharmonicity for bothharmonic and pseudoharmonic maps from either a compact Sasakian manifold or a completeSasakian manifold. Recall that Petit [17] gave similar results for harmonic maps froma compact Sasakian manifold by using tools of Spinorial geometry, although he didn’tmention the notion of CR pluirharmonicity. The moving frame method, which enables usto treat both cases of harmonic maps and pseudoharmonic maps, seems more closer to theclassical methods in differential geometry. Inspired by Sampson’s technique (cf. also [6]),we introduce θ W = ( φ iα φ i ¯ αβ θ β + φ i ¯ α φ iα ¯ β θ ¯ β ) . (6.1)Note that θ W consists of partial terms of θ W . Lemma 6.1. divθ W = 2 | φ iα ¯ β | + φ iα φ iβ ¯ β ¯ α + φ i ¯ α φ i ¯ ββα − φ iα φ jβ φ k ¯ α φ l ¯ β ’ R ijkl −√− m − φ iα φ iβ A ¯ α ¯ β − φ i ¯ α φ i ¯ β A αβ ) − √− φ iα φ i α − φ i ¯ α φ i α ) . (6.2) Proof.
Since the computation for deriving (6.2) is similar to that in Lemma 4.1, we omitits details.
Theorem 6.1.
Let ( M, J, θ ) be a compact Sasakian manifold of dimension m +1 and ( N, h ) be a Riemannian manifold with nonpositive Hermitian curvature. Suppose φ : M → N iseither a harmonic map or a pseudoharmonic map. Then φ is CR pluriharmonic and φ iα φ jβ φ k ¯ α φ l ¯ β ’ R ijkl = 0 . (6.3) Proof.
Since N has a nonpositive Hermitian curvature, the sectional curvature is nonpos-itive. According to Proposition 5.1 and Theorem 5.1, we know that the conditon that φ is harmonic is equivalent to that φ is pseudoharmonic. Besides, the map is basic in thiscircumstance. By (3.7), we have φ iα ¯ β = φ i ¯ βα for any α, β . Then we obtain τ ( φ ) = 2 φ iβ ¯ β E i and φ i ¯ ββα = φ iβ ¯ βα . 19y (6.2) and the fact that M is Sasakian, we get divθ W = 2 | φ iα ¯ β | + φ iα φ iβ ¯ β ¯ α + φ i ¯ α φ i ¯ ββα − φ iα φ jβ φ k ¯ α φ l ¯ β ’ R ijkl = 2 | φ iα ¯ β | + 12 hh d b φ, ∇ b τ ( φ ) ii − φ iα φ jβ φ k ¯ α φ l ¯ β ’ R ijkl = 2 | φ iα ¯ β | − φ iα φ jβ φ k ¯ α φ l ¯ β ’ R ijkl . (6.4)Since N has nonpositive Hermitian curvature, we have φ iα φ jβ φ k ¯ α φ l ¯ β ’ R ijkl ≤ . By the divergence theorem, we derive from (6.4) that φ is a CR pluriharmonic map withproperty (6.3).Let ( N n , h ) be a K¨ahler manifold. The curvature operator Q of N is defined by h Q ( X ∧ Y ) , Z ∧ W i = h R ( X, Y ) W, Z i for any X, Y, Z, W ∈ T M . The complex extension of Q to ∧ T C N is also denoted by Q .We introduce ≪ Q ( X ∧ Y ) , Z ∧ W ≫ = h Q ( X ∧ Y ) , Z ∧ W i . The K¨ahler identity of N yields Q | ∧ (2 , T C N = Q | ∧ (0 , T C N = 0 . Set Q (1 , = Q : ∧ (1 , T C N → ∧ (1 , T C N. Definition 6.1. (cf. [22]) Let ( N n , h ) be a K¨ahler manifold. The curvature tensor of ( N, h ) is said to be strongly negative (resp. strongly semi-negative) if ≪ Q (1 , ( ξ ) , ξ ≫ = h Q (1 , ( ξ ) , ξ i < (resp. ≤ )for any ξ = ( Z ∧ W ) (1 , = 0 , Z, W ∈ Γ ∞ ( T N C ) . Remark 6.1.
By comparing the Definitions 4.1 and 6.1, we find that the notions of non-positive Hermitian curvature and strongly semi-negative curvature are equivalent for K¨ahlermanifolds. However, we should point out that one cannot introduce the notion of negativeHermitian curvature for K¨ahler manifolds due to the K¨ahler identity.
Let θ W = φ ¯ iα φ i ¯ αβ θ β + φ i ¯ α φ ¯ iα ¯ β θ ¯ β . (6.5)Then we have divθ W = 2 | φ iα ¯ β | + φ ¯ iα φ iβ ¯ β ¯ α + φ i ¯ α φ ¯ i ¯ ββα − ≪ Q ( φ α ∧ φ β ) , φ α ∧ φ β ≫−√− m − φ ¯ iα φ iβ A ¯ α ¯ β − φ i ¯ α φ ¯ i ¯ β A αβ ) − √− φ ¯ iα φ i α − φ i ¯ α φ ¯ i α ) . (6.6)20 heorem 6.2. Let φ : ( M m +1 , J, θ ) → ( N, h ) be a harmonic or pseudoharmonic map froma compact Sasakian manifold into a K¨ahler manifold with strongly semi-negative curvature.Then φ is a CR pluriharmonic map and hh Q ( φ α ∧ φ β ) , φ α ∧ φ β ii = 0 , (6.7) where φ α = dφ ( T α ) .Proof. Since strongly semi-negative curvature implies non-positive sectional curvature, weget that φ must be pseudoharmonic and basic. Then we have φ iα ¯ β = φ i ¯ βα and φ i α = φ i α = 0.So we get τ ( φ ) = 2( φ iβ ¯ β E i + φ ¯ iβ ¯ β E ¯ i ) = 0, i.e., φ iβ ¯ β = φ ¯ iβ ¯ β = 0. As M is Sasakian, by (6.6)we have divθ W = 2 | φ iα ¯ β | − ≪ Q ( φ α ∧ φ β ) , φ α ∧ φ β ≫ . (6.8)The divergence theorem implies φ is CR pluriharmonic and hh Q ( φ α ∧ φ β ) , φ α ∧ φ β ii = 0.Now we attempt to give some conditions to ensure CR pluriharmonicity for harmonicand pseudoharmonic maps from complete noncompact Sasakian manifolds. Theorem 6.3.
Let ( M, J, θ ) be a complete noncompact Sasakian manifold and ( N, h ) be aRiemannian manifold with nonpositive Hermitian curvature. Suppose φ : M → N is eithera harmonic map or a pseudoharmonic map. If φ satisfies ( Z ∂B r e ( φ ) dS ) − / ∈ L (+ ∞ ) , (6.9) then φ is a CR pluriharmonic map with the property (6.3).Proof. By Theorem 5.2, we get that φ is basic. Under the conditions in the theorem, by(6.2) we have divθ W ≥ | φ iα ¯ β | . Using the divergence theorem, we get Z ∂B r θ W ( ∂∂r ) dS ≥ Z B r | φ iα ¯ β | Ψ . (6.10)On the other hand, by the definition of θ W , we have Z ∂B r θ W ( ∂∂r ) dS ≤ { Z ∂B r e H ( φ ) dS } { Z ∂B r | φ iα ¯ β | Ψ } . (6.11)Putting together (6.10) and (6.11) and squaring we finally get γ ( r ) ≤ ( Z ∂B r e H ( φ ) dS ) γ ′ ( r ) , (6.12)where we have set γ ( r ) = Z B r | φ iα ¯ β | Ψ . φ is not CR pluriharmonic. Then there exists a R > γ ( R ) >
0. For any r ≥ R , from (6.12) we can deduce γ ( R ) − − γ ( r ) − ≥ Z rR dt R ∂B t e H ( φ ) , and letting r → + ∞ we contradict (6.9). Hence φ is CR pluriharmonic. By definition, wehave θ W ≡
0. Then (6.2) implies that φ satisfies (6.3). Corollary 6.1.
Let ( M, J, θ ) be a complete noncompact Sasakian manifold and ( N, h ) be aRiemannian manifold with nonpositive Hermitian curvature. Suppose φ : M → N is eithera harmonic map or a pseudoharmonic map. If φ satisfies Z B r e ( φ )Ψ ≤ Cr , then φ is a CR pluriharmonic map with the property (6.3). Theorem 6.4.
Let φ : ( M m +1 , J, θ ) → ( N, h ) be a harmonic or pseudoharmonic mapfrom a complete noncompact Sasakian manifold into a K¨ahler manifold with strongly semi-negative curvature. If φ satisfies ( Z ∂B r e ( φ ) dS ) − / ∈ L (+ ∞ ) , (6.13) then φ is a CR pluriharmonic map with the property (6.7).Proof. Obviously, the map φ is basic, and hence φ iα ¯ β = φ i ¯ βα . It follows from (6.8) and thedivergence that 2 Z B r | φ iα ¯ β | Ψ ≤ Z B r divθ W Ψ = Z ∂B r θ W ( ∂∂r ) dS. By the definition of θ W , we have Z ∂B r θ W ( ∂∂r ) dS ≤ { Z ∂B r | φ i ¯ α | dS } / { Z ∂B r | φ i ¯ αβ | dS } / . Set ρ ( r ) = Z B r | φ i ¯ αβ | Ψ . Then ρ ( r ) ≤ ρ ′ ( r )( Z ∂B r | φ i ¯ α | dS ) . (6.14)Suppose that φ isn’t CR pluriharmonic, then there exists a R > ρ ( r ) > r > R . Fix such a R . From (6.14) we deduce the following ρ ( R ) − − ρ ( r ) − ≥ Z rR dt R ∂B r | φ i ¯ α | , and letting r → + ∞ we contradict (6.13). Hence φ is CR pluriharmonic. By definition, weget that θ W ≡
0. Then (6.6) implies that φ satisfies (6.7).22 orollary 6.2. Let φ : ( M m +1 , J, θ ) → ( N, h ) be a harmonic or pseudoharmonic mapfrom a complete noncompact Sasakian manifold into a K¨ahler manifold with strongly semi-negative curvature. If φ satisfies Z B r e ( φ )Ψ ≤ Cr , then φ is a CR pluriharmonic map with the property (6.7). In this section, we will establish some results of Siu-Sampson type for both harmonicmaps and pseudoharmonic maps from compact Sasakian manifolds. Similar to the resultsfor harmonic maps from K¨ahler manifolds in [5, 21, 22], we may derive CR holomorphicityunder rank conditions for harmonic and pseudoharmonic maps from compact Sasakianmanifolds by analysing the curvature equations (6.7). Note that Petit [17] also gave the CRholomorphicity results for harmonic maps from Sasakian manifolds using spinorial geometry.As mentioned previously, our method is different from his. Besides recapturing Petit’sresults by using the moving frame method, we also add some new results which includethe results for pseudoharmonic maps, the conic extension of harmonic maps from Sasakianmanifolds and a unique continuation theorem for CR holomorphicity.Suppose now that the target manifold N is a locally symmetric space of noncompacttype. Then the universal covering manifold of N is a symmetric space G/K, where K isa connected and closed subgroup of the noncompact connected Lie group G , and G/K isgiven the invariant metric determined by the Killing form h , i on g . If the correspondingCartan decomposition of the Lie algebra of G is g = k + p , then the real tangent space of N at any point can be identified with p . The curvature tensor of N is given by˜ R ( X, Y ) Z = − [[ X, Y ] , Z ] , for any X, Y, Z ∈ p , and the Hermitian curvature of N is given by h ˜ R ( X, Y ) Y , X i = h [ X, Y ] , [ X, Y ] i . (7.1)Therefore, (6.3) yields that [ dφ ( T α ) , dφ ( T β )] = 0 , (7.2)for any α, β . In this way, we get Proposition 7.1.
Let ( M, J, θ ) be a compact Sasakian manifold and N a locally symmetricspace of noncompact type. If φ : M → N is either a harmonic map or a pseudoharmonicmap, then φ is CR pluriharmonic and for any x ∈ M , dφ x maps T , M x onto an abeliansubspace W of p ⊗ C . Under the assumption of Proposition 7.1, the image under dφ x of real tangent space T x M is the subspace of real points of space W + W ⊂ T C φ ( x ) N , so that dim R dφ x ( T x M ) = dim C ( W + W ) ≤ dim C W. rank R ( dφ ) ≤ max { dim C W | W ⊂ p ⊗ C , [ W, W ] = 0 } . (7.3)When G = SO (1 , n ), then dimW ≤ Corollary 7.1.
Let ( M, J, θ ) be a compact Sasakian manifold and N a manifold of constantnegative curvature. If φ : M → N is harmonic or pseudoharmonic, then rank R ( dφ ) ≤ . If G/K is a Hermitian symmetric space, then corresponding to any invariant complexstructure on
G/K we have the decomposition p ⊗ C = p , ⊕ p , , and the integrability condition [ p , , p , ] ⊂ p , is equivalent, in view of [ p , p ] ⊂ k , to[ p , , p , ] = 0, thus p , is an abelian subalgebra of p ⊗ C . Lemma 7.1. (cf. [5]) Let
G/K be a symmetric space of non-compact type. Let W ⊂ p ⊗ C be an abelian subspace. Then dimW ≤ dim p ⊗ C . Equality holds in this inequality if andonly if G/K is Hermitian symmetric and W = p , for any invariant complex structure on G/K . From (7.3) and Lemma 7.1, we get immediately the following result.
Corollary 7.2.
Let φ : M → N be as in Proposition 7.1 and suppose that N is not locallyHermitian symmetric. Then rankdφ < dimN . The above corollary use only the case of strict inequality in Lemma 7.1. We have treatedthe case of equality in such detail in order to obtain the following theorem.
Theorem 7.1.
Let ( M, J, θ ) be a compact Sasakian manifold and N a locally Hermitiansymmetric space of noncompact type whose universal cover does not contain the hyperbolicplane as a factor. If φ : M → N is either a harmonic map or a pseudoharmonic map, andthere is a point x ∈ M such that dφ ( T x M ) = T φ ( x ) N , then φ is CR holomorphic.Proof. Since dφ ( T , M ) is an abelian subspace of half the dimension, it must be p , foran invariant complex structure on N , i.e., dφ x ( T , M x ) = p , . Consequently this propertymust hold on a neighborhood U of x . By Proposition 7.1 and Proposition 2.4, we have dφ ( T ) = 0. Therefore, the map φ is CR holomorphic on U . We get that the map φ is CRholomorphic on M by the following unique continuation Proposition 7.3.Now, we will give some fundamental knowledge about the warped product. Let ( B, g B )and ( S, g S ) be two Riemannian manifolds and f be a positive smooth function on B . Con-sider the product manifold B × S with its natural projections π B : B × S → B and π S : B × S → S . The warped product B × f S is the manifold B × S furnished with thefollowing Riemannian metric ˜ g = π ∗ B ( g B ) + ( f ◦ π B ) π ∗ S ( g S ) . (7.4)The Levi-Civita connection of N = B × f S can now be related to those of B and S asfollows. 24 emma 7.2. (cf. [16, p. 206]) Let ˜ ∇ , B ∇ and S ∇ be the Levi-Civita connections on N , B and S respectively. If X , Y are vector fields on S and V , W are vector fields on B , the liftof X, Y, V, W to B × f S is also denoted by the same notations, then(i) ‹ ∇ V W is the lift of B ∇ V W (ii) ‹ ∇ V X = ‹ ∇ X V = V ff X ;(iii) ( ‹ ∇ X Y ) B = − (˜ g ( X, Y ) /f ) gradf ;(iv) ( ‹ ∇ X Y ) S is the lift of S ∇ X Y on S . Now we consider the special case: let (
M, θ, J ) be a strictly pseudoconvex CR manifoldand C ( M ) be the manifold R + × r M endowed with the metric ˜ g = dr + r g θ . Therefore,by Lemma 7.2, we have ‹ ∇ ∂∂r ∂∂r = 0 , ‹ ∇ ∂∂r X = ‹ ∇ X ∂∂r = 1 r X, ‹ ∇ X Y = ∇ θX Y − g θ ( X, Y ) r ∂∂r . (7.5) Proposition 7.2. (cf. [2]) If ( M, J, θ ) is a Sasakian manifold, then ( C ( M ) , ˜ g ) is K¨ahler.Proof. Set ζ = r ∂∂r and define smooth section of End T C ( M ) by the formula˜ J Y = J Y − θ ( Y ) ζ, ˜ J ζ = T. (7.6)It is easy to see that ˜ J is an almost complex structure on C ( M ) and the metric ˜ g isHermitian. From (7.5) and (7.6) we can show that ‹ ∇ ˜ J = 0. Thus C ( M ) is K¨ahler.By (2.4), (7.5) and (7.6), we can derive the following Lemmas 7.3, 7.4 and 7.5. Lemma 7.3.
Let ( M m +1 , J, θ ) be a Sasakian manifold, ( C ( M ) , ˜ g ) its cone manifold, ( N n , h ) a Riemannian manifold. If φ : M → N is a harmonic map, then the conic ex-tension ˜ φ : C ( M ) → N defined by ˜ φ ( x, r ) = φ ( x ) (7.7) is also harmonic. Lemma 7.4.
Let ( M m +1 , J, θ ) be a Sasakian manifold, ( C ( M ) , ˜ g ) its cone manifold, ( N, h ) a Riemannian manifold. If φ : M → N is a CR pluriharmonic map, then the conic extension ˜ φ is a pluriharmonic map. Lemma 7.5.
Let φ : ( M, J, θ ) → ( N, h, J ′ ) be a smooth map from a Sasakian manifold toa K¨ahler manifold, ( C ( M ) , ˜ g ) the cone manifold of M , the conic extension of φ is definedby (7.7). Then φ is a CR holomorphic (resp. CR anti-holomorphic) map if and only if ˜ φ is holomorphic (resp. anti-holomorphic). In [22], Siu derived the following unique continuation theorem for holomorphicity.
Lemma 7.6. (cf. [22]) Suppose
M, N are two K¨ahler manifolds and φ : M → N is aharmonic map. Let U be a nonempty open subset of M . If φ is holomorphic (resp. anti-holomorphic) on U , then φ is holomorphic (resp. anti-holomorphic) on M . From the Lemmas 7.3, 7.5 and 7.6, we get the following unique continuation theorem.25 roposition 7.3.
Let φ : ( M m +1 , J, θ ) → ( N, h ) be a harmonic map from a connectedSasakian manifold to a K¨ahler manifold. Let U be a nonempty open subset of M . If φ isCR holomorphic (resp. CR anti-holomorphic) on U , then φ is CR holomorphic (resp. CRanti-holomorphic ) on M .Proof. From Lemma 7.3, we know that ˜ φ : C ( M ) → N is harmonic. Suppose φ is CRholomorphic on U . It follows from Lemma 7.5 that ˜ φ is holomorphic on R + × r U . UsingLemmas 7.5 and 7.6, we conclude that φ is CR holomorphic on M .Now we may establish the following results. Theorem 7.2.
Let ( M m +1 , J, θ ) be a compact Sasakian manifold and N be a K¨ahlermanifold with strongly negative curvature. Suppose φ : M → N is either a harmonic map ora pseudoharmonic map, and rank R dφ ≥ at some point of M , then φ is CR holomorphicor CR anti-holomorphic on M .Proof. From Theorem 6.2 and Lemma 7.3, we know that ˜ φ is harmonic. By Siu’s results,we have ˜ φ is ± holomorphic on C ( M ). By Proposition 7.3, we conclude that φ is CR ± holomorphic on M .Keeping in mind Udagawa’s proof to Theorem 4 of [26] the following result is relevant. Theorem 7.3.
Every CR pluriharmonic map φ : ( M, J, θ ) → ( N, h ) from a Sasakianmanifold M into an irreducible Hermitian symmetric space N of compact or noncompacttype is CR ± holomorphic if M ax M rank R dφ ≥ P ( N ) + 1 , where P ( N ) is the degree ofstrong non-degenerate of the bisectional curvature of N (cf. [23] for the definition of thedegree of strong non-degenerate of the bisectional curvature of N ).Proof. By Lemma 7.4, we have ˜ φ is pluriharmonic. Since M ax M rank R dφ ≥ P ( N ) +1 implies that M ax C ( M ) rank R d ˜ φ ≥ P ( N ) + 1, by Theorem 4 of [26] we get that ˜ φ is ± holomorphic. From Lemma 7.5, we prove that φ is CR ± holomorphic. Acknowledgments
This work was partially supported by the National Natural Science Foundation of China[grant number 11271071] and Laboratory of Mathematics for Nonlinear Science, Fudan;research of the last author was partially supported by HUST Innovation Research Grant0118011034.
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