Pseudo-Harmonic Maps From Complete Noncompact Pseudo ...

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Feb 22, 2018 - problem was settled by Hamilton [16] under the nonpositive condition of the targets. Li-Tam. [23] deduce ...... [16] R.S. Hamilton. Harmonic maps ...
arXiv:1802.08034v1 [math.DG] 22 Feb 2018

Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds To Regular Balls Tian Chong

Yuxin Dong∗

Yibin Ren†

Wei Zhang

February 23, 2018

Abstract In this paper, we give an estimate of sub-Laplacian of Riemannian distance functions in pseudo-Hermitian manifolds which plays a similar role as Laplacian comparison theorem, and deduce a prior horizontal gradient estimate of pseudo-harmonic maps from pseudo-Hermitian manifolds to regular balls of Riemannian manifolds. As an application, Liouville theorem will be established under the conditions of nonnegative pseudo-Hermitian Ricci curvature and vanishing pseudo-Hermitian torsion. Moreover, we obtain the existence of pseudo-harmonic maps from complete noncompact pseudoHermitian manifolds to regular balls of Riemannian manifolds.

1

Introduction

Let (M, θ) be a pseudo-Hermitian manifold and (N, h) be a Riemannian manifold. The horizontal energy of a smooth map f : M → N is Z EH (f ) = |db f |2 dV M

where db f is the horizontal part of df . Its Euler-Lagrange equation of EH is given by X ∆ Ξi (f ) = ∆b f i + Γijk (f )hdbf i , db f k i = 0 j,k

in local coordinates where Γijk ’s are Christoffel symbols of Levi-Civita connection in (N, h). By definition, the pseudo-harmonic map is an analogue of harmonic map in pseudoHermitian geometry. For harmonic maps, Eells and Sampson proved the existence theorem from closed Riemannian manifolds to nonpositively curved target manifolds. The Dirichlet Keywords: Sub-Laplacian Comparison Theorem, Regular Ball, Pseudo-Harmonic Maps, Horizontal Gradient Estimate, Liouville Theorem, Existence Theorem MSC 2010: 58E20, 53C25, 32V05 ∗ Supported by NSFC grant No. 11771087, and LMNS, Fudan. † Corresponding author.

1

problem was settled by Hamilton [16] under the nonpositive condition of the targets. Li-Tam [23] deduce the existence of harmonic maps from complete noncompact Riemannian manifold via heat flow and then Ding-Wang [11] improved it by elliptic method. Later, Li-Wang [21, 22] studied harmonic heat flow on complete noncompact Riemannian manifold and found some harmonic map when the target has either nonpositive or positive sectional curvature. Recently, Chen-Jost-Wang [8] applied maximum principle and continuity method to derive the existence of Dirichlet V-harmonic maps. Chen, Jost-Qiu [7] found a prior gradient estimate of V-harmonic maps and then obtained the existence and Liouville theorem of Vharmonic maps on complete noncompact Riemannian manifolds. There are also many other important researches about the existence theorem and Liouville theorem of harmonic maps, such as [9, 10, 17, 19, 24]. For pseudo-harmonic maps, the authors in [5, 25] obtained the similar version of Eells-Sampson existence theorem. When the target has positive sectional curvature, Jost-Xu [18] developed Hildebrandt-Kaul-Widman’s method (cf. [17]) and solved the Dirichlet problem of pseudo-harmonic maps under some convexity condition. As we know, Laplacian comparison theorem is an important tool in Riemannian geometry. Its extension to Sasakian geometry has been studied in [1, 3, 6, 20]. However, up to now, there is no satisfactory comparison theorem for a pseudo-Hermitian manifold, which is not Sasakian. For our purpose, we will give a new sub-Laplacian comparison theorem for a pseudo-Hermitian manifold. Note that the Riemannian distance associated with Webster metric has better regularity than the Carnot-Carath´eodory-distance, and its variational theory is well studied in Riemannian geometry. By the index comparison theorem in Riemannian geometry, we can derive an estimate of sub-Laplacian of Riemannian distance on pseudo-Hermitian manifolds (Section 3). As an application, we will give a prior horizontal gradient estimate of pseudo-harmonic maps from complete noncompact pseudo-Hermitian manifolds to regular balls of Riemannian manifolds (Section 4) and then Liouville theorem will be established. Based on Jost-Xu’s result [18], the Dirichlet pseudo-harmonic maps will approach some pseudo-harmonic maps on complete noncompact pseudo-Hermitian manifolds when the targets are regular balls (Section 5).

2

Basic Notions

In this section, we present some basic notions of pseudo-Hermitian geometry and pseudoharmonic maps. For details, the readers may refer to [13, 30]. Recall that a smooth manifold M of real dimension (2n + 1) is said to be a CR manifold if there exists a smooth rank n complex subbundle T1,0 M ⊂ T M ⊗ C such that T1,0 M ∩ T0,1 M = {0} [Γ(T1,0 M), Γ(T1,0 M)] ⊂ Γ(T1,0 M)

(2.1) (2.2)

where T0,1 M = T1,0 M is the complex conjugate of T1,0 M. Equivalently, the CR structure may also be described by the real subbundle HM = Re {T1,0 M ⊕ T0,1 M} of T M which carries an almost complex structure J : HM → HM defined by J(X + X) = i(X − X) for any X ∈ T1,0 M. Since HM is naturally oriented by the almost complex structure J, then M is orientable if and only if there exists a global nowhere vanishing 1-form θ such that 2

HM = Ker(θ). Any such section θ is referred to as a pseudo-Hermitian structure on M. The space of all pseudo-Hermitian structure is 1-dimensional. The Levi form Lθ of a given pseudo-Hermitian structure θ is defined by Lθ (X, Y ) = dθ(X, JY ) for any X, Y ∈ HM. An orientable CR manifold (M, HM, J) is called strictly pseudo-convex if Lθ is positive definite for some θ. When (M, HM, J) is strictly pseudo-convex, there exists a pseudo-Hermitian structure θ such that Lθ is positive. The quadruple (M, HM, J, θ) is called a pseudo-Hermitian manifold. For simplicity, we denote it by (M, θ). This paper is discussed in the pseudo-Hermitian manifolds. For a pseudo-Hermitian manifold (M, θ), there exists a unique nowhere zero vector field ξ, called the Reeb vector field, transverse to HM and satisfying ξy θ = 1, ξy dθ = 0. There is a decomposition of the tangent bundle T M: T M = HM ⊕ Rξ

(2.3)

∗ which induces the projection πH : T M → HM. Set Gθ = πH Lθ . Since Lθ is a metric on HM, it is natural to define a Riemannian metric

g θ = Gθ + θ ⊗ θ

(2.4)

which makes HM and Rξ orthogonal. Such metric gθ is called Webster metric, also denoted by h·, ·i. In the terminology of foliation geometry, Rξ provides a one-dimensional Reeb foliation and HM is its horizontal distribution. By requiring Jξ = 0, the almost complex structure J can be extended to an endomorphism of T M. The integrable condition (2.2) guarantees that gθ is J-invariant. Clearly θ ∧ (dθ)n differs a constant with the volume form of gθ . Henceforth we will regard it as the volume form and always omit it or denote it by dV for simplicity. It is remarkable that (M, HM, Gθ ) could also be viewed as a sub-Riemannian manifold which satisfies the strong bracket generating hypothesis. The completeness is well settled under the Carnot-Carath´eorody distance (cf. [28]). By definition, this distance is larger than Riemannian distance associated with the Webster metric gθ which implies that sub-Riemannian completeness is stronger than Riemannian one. In this paper, a pseudoHermitian manifold (M, θ) is called complete if it is complete associated with the Webster metric gθ . On a pseudo-Hermitian manifold, there exists a canonical connection ∇ (called TanakaWebster connection, cf. [13]) preserving the horizontal distribution, CR structure and Webster metric. Moreover, its torsion satisfies T∇ (X, Y ) = 2dθ(X, Y )ξ and T∇ (ξ, JX) + JT∇ (ξ, X) = 0. The pseudo-Hermitian torsion, denoted by τ , is a symmetric and traceless tensor defined by τ (X) = T∇ (ξ, X) for any X ∈ T M (cf. [13]). Set A = gθ (τ (X), Y ) for any X, Y ∈ T M. A pseudo-Hermitian manifold is called Sasakian if τ ≡ 0. 3

Let (M, θ) be a pseudo-Hermitian manifold of real dimension 2m + 1. Let {η1 , . . . , ηm } be a local unitary frame of T1,0 M defined on an open set U ⊂ M whose dual coframe is denoted by {θ1 , . . . θm }. Then the structure equations are given by  dθ = 2iθα ∧ θα¯ ,     α  dθ = θβ ∧ θβα + Aα¯β¯θ ∧ θβ , β¯ α  θ + θ  α ¯ = 0, β    dθα = θγ ∧ θα + Πα β

β

γ

β

where θβα ’s are the Tanaka-Webster connection 1-forms with respect to {ηα }. Webster [30] showed that α λ µ ¯ α µ ¯ α µ Παβ = 2i(θα ∧ τβ + θβ ∧ τ α ) + Rβλ¯ µ θ ∧ θ + Aµ ¯,β θ ∧ θ − Aµβ, θ ∧ θ α where Rβλ¯ µ is called the Webster curvature. He also derived the first Bianchi identity, i.e. Rαβλ¯ = R ¯ µ αλβ ¯ µ ¯ . So the pseudo-Hermitian Ricci curvature can be defined by Rλ¯ µ = Rααλ¯ ¯ µ . Tanaka [29] defined and then the pseudo-Hermitian scalar curvature is s = Rαα¯ = Rββα ¯ α ¯ the pseudo-Hermitian Ricci tensor R∗ by

R∗ X = −i

m X

R(ηλ , ηλ¯ )JX.

(2.5)

λ=1

It is obvious that R∗ ηα = Rαβ¯ηβ and 1 s = traceGθ R∗ . 2 Assume that (N, h) is a Riemannian manifold. Let {σ i } be an orthonormal frame of T ∗ N. Denote the Levi-Civita connection and the Riemannian curvature of (N, h) by ∇N and RN respectively. Suppose that f : M → N is a smooth map. The pullback connection on the pullback bundle f ∗ (T N) and the Tanaka-Webster connection induce a connection on T M ⊗ f ∗ (T N), also denoted by ∇. Definition 2.1. A smooth map f : M → N is called pseudo-harmonic if the tensor field ∆

Ξ(f ) = traceGθ ∇df |HM ×HM ≡ 0. Actually, pseudo-harmonic maps are the Dirichlet critical points of the horizontal energy (cf. [2, 13]) Z 1 EH (f ) = |db f |2 θ ∧ (dθ)m (2.6) 2 M where db f is the horizontal restriction of df . The sub-Laplacian ∆b u of a smooth function u is defined by ∆b u = traceGθ ∇b db u, which is viewed as the special case of Ξ acting on functions. 4

(2.7)

Lemma 2.2 (CR Bochner Formulas, cf. [5, 15, 26]). For any smooth map f : M → N, we have 1 i ∆b |db f |2 =|∇b db f |2 + h∇b Ξ(f ), dbf i + 4i(fα¯i f0α − fαi f0iα¯ ) 2 + 2Rαβ¯fα¯i fβi − 2i(n − 2)(fαi fβi Aα¯ β¯ − fα¯i fβ¯i Aαβ ) N N + 2(fα¯i fβj fβ¯k fαl Rijkl + fαi fβj fβ¯k fα¯l Rijkl )

1 N ∆b |f0 |2 =|∇b f0 |2 + h∇ξ Ξ(f ), f0 i + 2f0i fαj fα¯k f0l Rijkl 2 i i i i i i + 2(f0i fβi Aβ¯α,α ¯ + f0 fβ¯α ¯ + f0 fβ¯ Aβα,α ¯) ¯ Aβα + f0 fβα Aβ¯α

(2.8)

(2.9)

i where fAi and fAB are the components of df and ∇df respectively under the orthonormal α α coframe {θ, θ , θ ¯ } of T ∗ M and an orthonormal frame {σi } of T ∗ N, and f0 = df (ξ).

Let π(1,1) ∇b db f be the (1, 1)-part of ∇b df and ⊥ π(1,1) ∇b df = ∇b db f − π(1,1) ∇b db f

which is orthogonal to π(1,1) ∇b db f . The commutation relation (cf. [5, 15, 26]) i i fαi β¯ − fβα ¯ = 2if0 δαβ¯

shows that 2

|π(1,1) ∇b db f | ≥2 = ≥

1 2 1 2

m X

i fαi α¯ fαα ¯

α=1 m X α=1 m X α=1

 i  i 2 i i 2 |fαα¯ + fαα | + |f − f | ¯ αα ¯ αα ¯ i 2 |fαi α¯ − fαα ¯ |

=2m|f0 |2 .

(2.10)

Combining with Lemma 2.2, we have the following lemma. Lemma 2.3. Suppose that (M 2m+1 , θ) is a pseudo-Hermitian manifold with R∗ ≥ −k, and |A|, |div A| ≤ k1

(2.11)

and (N, h) is a Riemannian manifold with sectional curvature KN ≤ κ

(2.12)

for k, k1 , κ ≥ 0. Then there exists C1 = C1 (k, k1 ) such that for any pseudo-harmonic map f : M → N, we have ⊥ ∆b |db f |2 ≥(2 − ǫ)|∇b db f |2 + 2mǫ|f0 |2 + ǫ|π(1,1) ∇b d b f | 2

2 4 − ǫ1 |∇b f0 |2 − (C1 + 16ǫ−1 1 )|db f | − 2κ|db f |

5

(2.13)

and ⊥ ∆b |f0 |2 ≥ 2|∇bf0 |2 − 2κ|f0 |2 |db f |2 − C1 |π(1,1) ∇b db f |2 − C1 |f0 |2 − C1 |db f |2

(2.14)

where ǫ and ǫ1 are any positive number. In particular, if k = 0 and k1 = 0, then C1 = 0. Proof. For (2.13), due to (2.10) and the identity i i(fα¯i f0α − fαi f0iα¯ ) = −h∇b f0 , dbf ◦ Ji,

it suffice to prove that 1 N N fα¯i fβj fβ¯k fαl Rijkl + fαi fβj fβ¯k fα¯l Rijkl ≥ − κ|dbf |4 . 2

(2.15)

Set df (ηα ) = tα + it′α . Hence due to sectional curvature K N ≤ κ, a direct calculation shows that N N fα¯i fβj fβ¯k fαl Rijkl + fαi fβj fβ¯k fα¯l Rijkl

= 2 hRN (tβ , tα )tβ , tα i + hRN (tβ , t′α )tβ , t′α i + hRN (t′β , tα )t′β , tα i + hRN (t′β , t′α )t′β , t′α i m X (|tα |2 |tβ |2 + |t′α |2 |tβ |2 + |tα |2 |t′β |2 + |t′α |2 |t′β |2 ) ≥ −2κ α,β=1

= −2κ

m X α=1

(|tα |2 + |t′α |2 )

!

m X β=1

(|tβ |2 + |t′β |2 )

!

which, combining with |db f |2 = 2

m X α=1

hdf (ηα), df (ηα¯ )i = 2

m X α=1

htα + it′α , tα − it′α i = 2

m X α=1

(|tα |2 + |t′α |2 )

yields (2.15). Similarly, (2.14) follows from the following process N f0i fαj fα¯k f0l Rijkl = hRN (tα − it′α , f0 )(tα + it′α ), f0 i

= hRN (tα , f0 )tα , f0 i + hRN (t′α , f0 )t′α , f0 i ! m X (|tα |2 + |t′α |2 ) ≥ −κ|f0 |2 α=1

1 = − κ|f0 |2 |db f |2 . 2

6



At the end of this Section, we briefly recall Folland-Stein space. Let (M, θ) be a pseudoHermitian manifold and Ω ⋐ M. For any k ∈ N and p > 1, the Folland-Stein space Skp (Ω) is given by  Skp (Ω) = u ∈ Lp (Ω) ∇lb u ∈ Lp (Ω), l = 0, 1, . . . , k where ∇lb u is the horizontal restriction of ∇l u and its Skp -norm is defined by ||u||Skp(Ω) =

k X l=0

||∇lb u||Lp (Ω) .

Under this generalized Sobolev space, the interior regularity theorem of subelliptic equations will behave as elliptic ones. Theorem 2.4 (Theorem 3.17 in [13], Theorem 16 in [27]). Suppose that (M, θ) is a pseudoHermitian manifold and Ω ⋐ M. Assume that u, v ∈ L1loc (Ω) and ∆b u = v in the distribution p sense. For any χ ∈ C0∞ (Ω), if v ∈ Skp (Ω) with p > 1 and k ∈ N, then χu ∈ Sk+2 (Ω) and   p p p (2.16) + ||v|| ||u|| ≤ C ||χu||Sk+2 χ L (Ω) Sk (Ω) (Ω) where Cχ only depends on χ.

Due to the commutation relation (cf. [26]) uαβ¯ − uβα ¯ = 2iu0 δαβ¯, we find that Reeb derivatives can be controlled by horizontal derivatives with double times. Hence Folland-Stein space may be embedded into some classical Sobolev space. Theorem 2.5 (Theorem 19.1 in [14]). Suppose that (M, θ) is a pseudo-Hermitian manifold and Ω ⋐ M. Then for any k ∈ N and p > 1, Skp (Ω) ⊂ Lpk/2 (Ω) which is the classical Sobolev space. Moreover, for any r ∈ N and p > dim M, there exists k ∈ N such that Skp (Ω) ⊂ C r,α (Ω).

3

CR Comparison Theorem

This section will estimate the lower bound of sub-Laplacian of Riemannian distance function under some conditions of pseudo-Hermitian Ricci curvature and pseudo-Hermitian torsion, which plays a similar role as Laplacian comparison theorem in Riemannian geometry. Suppose that (M 2m+1 , θ) is a complete noncompact pseudo-Hermitian manifold. Let r be the Riemannian distance from a fixed point x0 ∈ M. We formulate all Riemannian 7

symbols with “ ˆ ” to distinguish with ones in pseudo-Hermitian geometry, such as Leviˆ and Riemannian curvature tensor R. ˆ Lemma 1.3 in [13] shows the Civita connection ∇ relation of Tanaka-Webster connection and Levi-Civita connection associated with Webster metric: ˆ = ∇ − (dθ + A) ⊗ ξ + τ ⊗ θ + 2θ ⊙ J ∇

(3.1)

where 2θ ⊙ J = θ ⊗ J + J ⊗ θ. Hence the sub-Laplacian of r can also be calculated by Levi-Civita connection as follows: d ∆b r = traceGθ Hess(r) (3.2) HM ×HM [ is the Riemannian Hessian. where Hess Let’s recall the Index Lemma in Riemannian geometry (cf. [12] in page 212).

Lemma 3.1 (Index Lemma). Let γ : [0, a] → M be a Riemannian geodesic with no conjugate and X be a Jacobi field along γ with X ⊥ γ˙ and X(0) = 0. If V ∈ Γ(T M) γ with V (0) = 0, V (a) = X(a) and V ⊥ γ. ˙ Then Ia (X, X) ≤ Ia (V, V )

where Ia (V, V ) =

Z

a 0

(3.3)

  ˆ γ) ∇ ˆ γ˙ V 2 − hR(V, ˙ γ, ˙ V i dt

2m Now let γ : [0, a] → M be a Riemannian geodesic with no conjugate and {eB (a)}B=1 be a orthonormal basis of HM γ(a) . Set e⊥ B (a) = eB (a) − heB (a), ∇ri∇r ∈ T M γ(a)

[ which is perpendicular to γ(a) ˙ = ∇r γ(a) . Since Hess(r)(∇r, ·) = 0, then

2m 2m X X ⊥ ⊥ [ [ Hess(r)(e Hess(r)(eB (a), eB (a)) = ∆b r γ(a) = B (a), eB (a))

(3.4)

B=1

B=1

Using the Riemannian exponential map, we could extend e⊥ B (a) as a Jacobi field UB along γ with UB (0) = 0, UB (a) = e⊥ ˙ = 0. B (a), [UB , γ] Hence we find ⊥ ⊥ ˆ [ [ Hess(r)(e B (a), eB (a)) = Hess(r)(UB (a), UB (a)) = hUB (a), ∇UB (a) ∇ri Z a d ˆ ˆ ˆ γ˙ UB idt = hUB , ∇UB ∇ri γ(a) = hUB , ∇γ˙ UB i γ(a) = hUB , ∇ dt 0 Z a  2 ∇ ˆ γ˙ UB + hUB , ∇ ˆ γ˙ ∇ ˆ γ˙ UB i dt = Ia (UB , UB ), = 0

8

where the last equation is due to the Jacobi equation. Hence the sub-Laplacian of r can be rewritten as 2m X Ia (UB , UB ). ∆b r γ(a) =

(3.5)

B=1

Lemma 3.2. Let eB (t) be the parallel extension of eB (a) along γ with respect to TanakaWebster connection. Suppose the curvature along γ satisfies 2m X

ˆ B , ∇r)∇r, eB i ≥ −kˆ hR(e

(3.6)

B=1

and the pseudo-Hermitian torsion is bounded |A| ≤ k1 ,

(3.7)

ˆ k1 ≥ 0 Then there is a constant C2 = C2 (m) such that for some for k, q 1 ˆ ∆b r γ(a) ≤ C2 ( + 1 + k1 + k12 + k). a

(3.8)

Proof. Due to (3.1), we have

ˆ γ˙ eB = −[dθ(γ, ∇ ˙ eB ) + A(γ, ˙ eB )]ξ + θ(γ)Je ˙ ˙ eB ) + A(γ, ˙ eB )]ξ + θ(γ)Je ˙ B = −[gθ (J γ, B which implies that 2m 2m 2 X X   ˆ 2 |gθ (J γ, ˙ eB )|2 + 2gθ (J γ, ˙ eB )A(γ, ˙ eB ) + |A(γ, ˙ eB )|2 ˙ + ∇γ˙ eB = 2m |θ(γ)|

B=1

B=1

≤ 2m + 1 + 2A(γ, ˙ J γ) ˙ +

m X

B=1

|A(γ, ˙ eB )|2 ≤ 2m + 1 + 2k1 + k12

Set e′B (t) = eB (t) − heB (t), ∇ri∇r ⊥ γ, ˙

√ 1 sκ (t) = √ sinh( κt), κ

and VB (t) =

sκ (t) ′ e (t) sκ (a) B

where κ=

1 ˆ (6m + 3 + 6k1 + 3k12 + k). 3m

9

Hence VB (0) = 0, VB (a) = e′B (a), VB ⊥ γ˙ and 2 2 2 2m 2m 2m 2m 2 X X X sκ (t) s˙ κ (t) ′ 3 X s˙ κ (t) ′ sκ (t) ˆ ′ ˆ ′ ˆ ∇γ˙ VB = sκ (a) ∇γ˙ eB sκ (a) eB + 3 sκ (a) eB + sκ (a) ∇γ˙ eB ≤ 2 B=1 B=1 B=1 B=1 2 s˙ κ (t) 2 + 3(2m + 1 + 2k1 + k 2 ) sκ (t) ≤ 3m 1 sκ (a) sκ (a)

due to Cauchy inequality. The Index Lemma and (3.5) show that 2m 2m X X Ia (VB , VB ) = ∆b r γ(a) ≤ B=1

B=1

Z

a

0

  ˆ B , ∇r)∇r, VB i dt ˆ γ˙ VB 2 − hR(V ∇

2 ! a s˙ κ (t) 2 s (t) ˆ κ dt + (6m + 3 + 6k1 + 3k12 + k) = 3m sκ (a) sκ (a) 0 Z a  3m ≤ |s˙ κ (t)|2 + κ|sκ (t)|2 dt 2 |sκ (a)| 0 √ √ = 3m κ coth κa 1 √ ≤ 3m( + κ) a Z

which finishes the proof. Since the condition (3.6) is independent of the choice of horizontal orthonormal frame of {eB }2m B=1 , then it can be rewritten by pseudo-Hermitian data due to the relationship between ˆ and the curvature tensor R associated with Tanakathe Riemannian curvature tensor R Webster connection ∇ (cf. Theorem 1.6 in [13]): ˆ R(X, Y )Z =R(X, Y )Z + (LX ∧ LY )Z + 2dθ(X, Y )JZ − gθ (S(X, Y ), Z)ξ + θ(Z)S(X, Y ) − 2gθ (θ ∧ O(X, Y ), Z)ξ + 2θ(Z)(θ ∧ O)(X, Y )

(3.9)

where S(X, Y ) =(∇X τ )Y − (∇Y τ )X O =τ 2 + 2Jτ − I L =τ + J Here I is the identity, that is I(X) = X. Suppose {eB }2m B=1 is a local real orthonormal basis 1 √ of HM with eα+m = Jeα for α = 1, . . . m and ηα = 2 (eα − iJeα ). Lemma 3.3. For X, Y ∈ T M, we have 2m X

B=1

ˆ B , X)Y,eB i = hR(e

2m X

B=1

hR(eB , X)Y, eB i

− 3hπH X, πH Y i + hτ X, τ Y i + (2m − |τ |2 )θ(X)θ(Y ) + div τ (X)θ(Y ) (3.10) 10

Proof. By (3.9) and eB ∈ HM, we have 2m X

ˆ B , X)Y, eB i hR(e

B=1 2m X

=

hR(eB , X)Y, eB i +

B=1 2m X

+

B=1

2m X

h(LeB ∧ LX)Y, eB i +

B=1

θ(Y )hS(eB , X), eB i +

2m X

B=1

2m X

B=1

2dθ(eB , X)hJY, eB i

2θ(Y )h(θ ∧ O)(eB , X), eB i

(3.11)

Now we see each terms in the right side except the first one. Note that 2m X

h(LeB ∧ LX)Y, eB i =

B=1

2m X

hLeB , Y ihLX, eB i − hLX, Y ihLeB , eB i

(3.12)

B=1

On one hand, since LX is horizontal and hLeB , Y i = heB , τ Y i − heB , JY i, then we find 2m X

B=1

hLeB , Y ihLX, eB i =hLX, τ Y i − hLX, JY i =hτ X, τ Y i + hJX, τ Y i − hτ X, JY i − hJX, JY i =hτ X, τ Y i − hπH X, πH Y i.

(3.13)

Here the last equation is due to τ J + Jτ = 0. On the other hand, hLeB , eB i = traceGθ τ + traceGθ J = 0.

(3.14)

Substituting (3.13) and (3.14) into (3.12), the result is 2m X

B=1

h(LeB ∧ LX)Y, eB i = hτ X, τ Y i − hπH X, πH Y i.

(3.15)

For the third term in (3.11), we have 2m X

B=1

2dθ(eB , X)hJY, eB i =

2m X

B=1

−2heB , JXihJY, eB i = −2hJX, JY i = −2hπH X, πH Y i. (3.16)

For the fourth term in (3.11), by the formula of S, we have 2m X

B=1

hS(eB , X), eB i =

2m X

B=1

h(∇eB τ )X, eB i + 11

2m X

B=1

h(∇X τ )eB , eB i = div τ (X)

(3.17)

since τ is traceless. For the fifth term, by the definition of O, we have 2m X

B=1

2h(θ ∧ O)(eB , X), eB i = =

2m X

B=1 2m X

B=1

−hθ(X)O(eB ), eB i −θ(X)h(τ 2 + 2Jτ − I)(eB ), eB i

=θ(X)(2m − |τ |2 )

(3.18)

due to −

2m X

hJτ (eB ), eB i =

B=1

2m X

B=1

hτ JeB , eB i =

m X α=1

hτ Jeα , eα i + hτ J 2 eα , Jeα i = 0.

By substituting (3.15), (3.16), (3.17) and (3.18) to (3.11), we get (3.10). Tanaka [29] gives the first Bianchi identity of R is S (R(X, Y )Z) = 2S (dθ(X, Y )τ (Z)) .

(3.19)

where S stands for the cyclic sum with respect to X, Y, Z ∈ HM. Lemma 3.4. For any X, Y ∈ T M, we have hR∗ X, Y i =

2m X

hR(eB , πH X)πH Y, eB i − 2(m − 1)A(JX, Y ),

(3.20)

B=1

Proof. Since JX is horizontal, we can use the first Bianchi identity (3.19) and obtain −i

m X α=1

R(ηα , ηα¯ )JX − i

= −i

m X α=1

m X α=1

R(ηα¯ , JX)ηα − i

2dθ(ηα , ηα¯ )τ JX − i

= 2mτ JX − 2

m X α=1



m X α=1

m X

R(JX, ηα )ηα¯

α=1

2dθ(ηα¯ , JX)τ ηα − i

τ J hηα¯ , Xiηα + hX, ηα iηα¯



m X

2dθ(JX, ηα )τ ηα¯

α=1

= 2(m − 1)τ JX.

(3.21)

On the other hand, note that i

m X α=1

R(ηα¯ , JX)ηα + i

m X α=1

R(JX, ηα )ηα¯ = − i

m X

=−J =−J 12

R(JX, ηα¯ )ηα + i

α=1 2m X

B=1

R(JX, ηα )ηα¯

α=1

α=1

m X

m X

R(JX, ηα¯ )ηα + R(JX, ηα )ηα¯ R(JX, eB )eB

!

! (3.22)

Substituting (3.22) into (3.21), we obtain 2m X

hR∗ X, Y i =

hR(eB , JX)JY, eB i + 2(m − 1)A(JX, Y ).

B=1

By replacing X, Y by JX, JY , the proof is finished. For any Y ∈ HM, using (3.9), we have 2m X

ˆ B , ξ)Y, eB i = hR(e

B=1

2m X

hR(eB , ξ)Y, eB i

B=1

and 2m X

B=1

ˆ B , Y )ξ, eB i = hR(e

2m X

B=1

hS(eB , Y ), eB i = div τ (Y ).

Applying the symmetric property of Riemannian curvature, we get 2m X

hR(eB , ξ)Y, eB i = div τ (Y ).

(3.23)

B=1

Combing Lemma 3.3, Lemma 3.4 and (3.23), we obtain the following lemma Lemma 3.5. For any X, Y ∈ T M, we have 2m X

B=1

ˆ B , X)Y, eB i = hR∗ X, Y i + 2(m − 1)A(X, JY ) + hτ X, τ Y i − 3hπH X, πH Y i hR(e + (2m − |τ |2 )θ(X)θ(Y ) + div τ (X)θ(Y ) + div τ (Y )θ(X)

(3.24)

Using Lemma 3.2, we obtain the following sub-Laplacian estimate of Riemannian distance function. Theorem 3.6. Suppose (M 2m+1 , θ) is a complete pseudo-Hermitian manifold and BR (x0 ) is the geodesic ball of radius R centered at x0 . If R∗ ≥ −2mk, and |A|, |divA| ≤ k1 , then there exists C3 = C3 (m) such that   q 1 2 ∆b r ≤ C3 + 1 + k + k1 + k1 , r where Cut(x0 ) is the cut locus of x0 .

13

on BR (x0 ),

on BR (x0 ) \ Cut(x0 )

(3.25)

4

Horizontal Gradient Estimates and Liouville Theorem

Suppose that (M 2m+1 , θ) is a complete noncompact pseudo-Hermitian manifold. Let r be the Riemannian distance function from x0 ∈ M associated with the Webster metric gθ and BR be the geodesic ball of radius R centered at x0 . Assume that R∗ ≥ −2mk, and |A|, |divA| ≤ k1 ,

on B2R

for some R ≥ 1. Theorem 3.6 shows that 1 ∆b r ≤ C4 ( + 1), r

on B2R \ Cut(x0 ),

(4.1)

where C4 = C(k, k1 ). Choose a cut-off function ϕ ∈ C ∞ ([0, ∞)) such that 1 ϕ [0,1] = 1, ϕ [2,∞) = 0, −C5′ |ϕ| 2 ≤ ϕ′ ≤ 0.

By defining χ(r) = ϕ( Rr ), since R ≥ 1, we find |∇b χ|2 C5 ≤ 2, χ R

∆b χ ≥ −

C5 R

(4.2)

where C5 = C5 (k, k1 ). Suppose that (N, h) is a Riemannian manifold with sectional curvature KN ≤ κ for some κ ≥ 0. Denote the Riemannian distance function from p0 ∈ N by ρ. Let BD = BD (p0 ) be a regular ball of radius D around p0 , that is D < 2√π κ and BD lies inside the cut locus of p0 where 2√π κ = +∞ if κ = 0. Set φ(t) =

(

√ 1−cos( κt) , κ t2 , 2

κ>0 . κ=0

and ψ(q) = φ ◦ ρ(q). Obviously, φ is an increasing function and ψ is at least C 2 in the cut locus of p0 . Moreover, Hessian comparison theorem shows that √ (4.3) Hess ψ ≥ cos( κρ) · h. Lemma 4.1. For any 0 < D < depending on D such that

π √ , 2 κ

there exist ν ∈ [1, 2), b > φ(D) and δ > 0 only

√ cos( κt) ν − 2κ > δ, b − φ(t) 14

∀t ∈ [0, D]

(4.4)

Proof. For the case κ > 0, it suffices to find ν ∈ [1, 2) and b > φ(D) such that ν ν  φ(D) < b < inf + (1 − )s , s∈[0,φ(D)] 2κ 2

(4.5)

which is obvious due to φ(D) < κ1 . The case κ = 0 is obvious by choosing ν = 1. Assume that f : B2R (x0 ) ⊂ M → BD (p0 ) is a pseudo-harmonic map with f (x0 ) = p0 . By (4.3), we have the following estimate: Lemma 4.2. Let ν, b, δ be given in Lemma 4.1. Then ν

∆b ψ ◦ f − 2κ|dbf |2 ≥ δ|db f |2 b−ψ◦f

(4.6)

To estimate |dbf |2 , we consider the following auxiliary function Φµχ = |dbf |2 + µχ|f0 |2 where µ will be determined later. Lemma 4.3. Suppose µ and ǫ satisfy C1 µ ≤ ǫ ≤ 1. If χ(x) 6= 0 and Φµχ (x) 6= 0, then at x, we have ∆b Φµχ ≥

1 − ǫ |∇b Φµχ |2 − 2κ|db f |2 Φµχ 2 Φµχ  + 2mǫ − C1 µχ − 4ǫ−1 µχ−1 |∇b χ|2 + µ∆b χ |f0 |2   − C1 + C1 µχ + 16(ǫµχ)−1 |dbf |2

(4.7)

Proof. The estimate (2.14) gives that

∆b (χ|f0 |2 ) =χ∆b |f0 |2 + 2h∇bχ, ∇b |f0 |2 i + |f0 |2 ∆b χ

⊥ ≥2χ|∇b f0 |2 − 2κχ|f0 |2 |db f |2 − C1 χ|π(1,1) ∇b db f |2 − C1 χ|f0 |2 − C1 χ|db f |2

+ 4h∇b χ ⊗ f0 , ∇b f0 i + |f0 |2 ∆b χ.

(4.8)

Using (2.13) with ǫ1 = ǫµχ and (4.8), we have ∆b Φµχ =∆b (|db f |2 + µχ|f0 |2 ) ≥(2 − ǫ)(|∇b db f |2 + µχ|∇b f0 |2 ) + 4µh∇b χ ⊗ f0 , ∇b f0 i + 2mǫ|f0 |2 − C1 µχ|f0 |2 + µ∆b χ|f0 |2   − C1 + C1 µχ + 16(ǫµχ)−1 |dbf |2 − 2κΦµχ |db f |2 15

(4.9)

By Cauchy inequality, we have the following estimate |∇b Φµχ |2 = |∇b (|db f |2 + µχ|f0 |2 )|2 √ √ = |∇b hdb f + µχf0 ⊗ θ, db f + µχf0 ⊗ θi|2   2 √ √ √ ∇ χ b = 4 db f + µχf0 ⊗ θ, ∇b db f + µχ∇b f0 ⊗ θ + µ √ ⊗ f0 ⊗ θ 2 χ 2 2 √ √ √ ∇b χ ≤ 4 db f + µχf0 ⊗ θ · ∇b db f + µχ∇b f0 ⊗ θ + µ √ ⊗ f0 ⊗ θ 2 χ   µ|∇bχ|2 2 2 2 |f0 | + µh∇b f0 , ∇b χ ⊗ f0 i = 4Φµχ |∇b db f | + µχ|∇b f0 | + 4χ which, using Cauchy inequality again, implies that (2 − ǫ)(|∇b db f |2 + µχ|∇b f0 |2 ) + 4µh∇b χ ⊗ f0 , ∇b f0 i  ≥ (2 − 2ǫ) |∇b db f |2 + µχ|∇b f0 |2 + ǫµχ|∇b f0 |2 + 4µh∇b χ ⊗ f0 , ∇b f0 i 1 − ǫ |∇b Φµχ |2 1 − ǫ µ|∇b χ|2 ≥ − |f0 |2 + (2 + 2ǫ)µh∇b χ ⊗ f0 , ∇b f0 i + ǫµχ|∇b f0 |2 2 Φµχ 2 χ   2 1 − ǫ |∇b Φµχ | |∇b χ|2 1 − ǫ (1 + ǫ)2 ≥ µ − + |f0 |2 2 Φµχ 2 ǫ χ |∇b χ|2 1 − ǫ |∇b Φµχ |2 − 4ǫ−1 µ |f0 |2 (4.10) ≥ 2 Φµχ χ due to ǫ ≤ 1 and 1 − ǫ (1 + ǫ)2 1 − ǫ (1 + ǫ)2 + ≤ + = 2ǫ−1 + ǫ + 1 ≤ 4ǫ−1 . 2 ǫ ǫ ǫ Submitting (4.10) to (4.9), we finished the proof. Set Fµχ =

Φµχ (b − ψ ◦ f )ν

where ν ∈ [1, 2) and b are determined in Lemma 4.1. The ǫ in Lemma 4.3 will be chosen as ǫ=

1 1 2 − < −1 ≤1 ν 2 ν

(4.11)

and µ satisfy C1 µ ≤ ǫ.

16

(4.12)

Let x be a maximum point of χFµχ on B2R whose value is nonzero. Assume that r is smooth at x. Otherwise we can modify the distance function r as [9]. Hence at x, we have ∇b χ ∇b Φµχ ∇b (ψ ◦ f ) + +ν χ Φµχ b−ψ◦f 2 ∆b Φµχ |∇b Φµχ |2 ∆b χ |∇b χ| − − + 0 ≥ ∆b ln(χFµχ ) = χ χ2 Φµχ Φ2µχ |∇b (ψ ◦ f )|2 ∆b (ψ ◦ f ) +ν +ν b−ψ◦f (b − ψ ◦ f )2 0 = ∇b ln(χFµχ ) =

(4.13)

(4.14)

By (4.7), (4.14) becomes 0≥

∆b χ |∇b χ|2 1 + ǫ |∇b Φµχ |2 ∆b (ψ ◦ f ) |∇b (ψ ◦ f )|2 2 − 2κ|d f | + ν − − + ν b χ χ2 2 Φ2µχ b−ψ◦f (b − ψ ◦ f )2    |db f |2 |∇b χ|2 |f0 |2  + 2mǫ − C1 µχ + µ∆b χ − 4ǫ−1 µ − C1 + C1 µχ + 16(ǫµχ)−1 χ Φµχ Φµχ (4.15)

Using (4.13) and Cauchy inequality, we have at x 2 1 + ǫ ∇b χ ∇b (ψ ◦ f ) 1 + ǫ |∇b Φµχ |2 =− +ν − 2 Φ2µχ 2 χ b−ψ◦f 2 1+ǫ |∇bχ|2 1 + ǫ 2 |∇b (ψ ◦ f )| ≥− (1 + ǫ−1 ) − (1 + ǫ )ν 2 2 2 χ2 2 (b − ψ ◦ f )2

(4.16)

Due to the choice (4.11) of ǫ, we can take ǫ2 =

2 2−ν −1= >0 ν(1 + ǫ) 2+ν

such that ν=

1+ǫ (1 + ǫ2 )ν 2 2

1+ǫ 2+ν (1 + ǫ−1 2 ) = 2 ν(2 − ν)

and

(4.17)

Substituting (4.16), (4.3) to (4.15), we have at x   ∆b χ |∇b χ|2 2+ν ∆b ψ ◦ f 0≥ − 1+ +ν − 2κ|db f |2 2 χ ν(2 − ν) χ b−ψ◦f   2  |db f |2 |f0 |2  −1 |∇b χ| − C1 + C1 µχ + 16(ǫµχ)−1 + 2mǫ − C1 µχ + µ∆b χ − 4ǫ µ χ Φµχ Φµχ The estimates (4.2) and Lemma 4.2 yield that   2  µCν |f0 |2  Cν 2 −1 |db f | + δ|db f | + 2mǫ − C1 µχ − − C1 + C1 µχ + 16(ǫµχ) 0≥− χR R Φµχ Φµχ 17

where Cν only depends on ν and C5 . By definition of Φµχ , |f0 |2 = µ−1 χ−1 (Φµχ − |db f |2 ) which shows at x,   Cν 1 Cν −1 2mǫµ − C1 χ − + 0≥− χR χ R       |db f |2 1 Cν −1 −1 + δΦµχ − 2mǫµ − C1 χ − − C1 + C1 µχ + 16(ǫµχ) χ R Φµχ       |db f |2 1 2Cν −1 −1 −1 ≥ 2mǫµ − C1 − + δχΦµχ − 2mǫµ − C1 χ + C1 µ + 16(ǫµ) χ R χΦµχ (4.18) To make the first bracket of the last line in (4.18) nonnegative, we can choose sufficiently small µ such that 2mǫµ−1 = C1 +

2Cν R

which also makes (4.12) right. Hence (χΦµχ )(x) ≤ δ −1 C6 (C1 , ν, R).

(4.19)

where 2Cν C6 (C1 , ν, R) = 2C1 + + mC1 R



2 −1 ν



2Cν C1 + R

−1

32ν 2 + m(2 − ν)2

  2Cν C1 + R (4.20)

which implies max χFµχ ≤

B2R (x0 )

χΦµχ C6 (C1 , ν, R) (x) ≤ . ν (b − ψ ◦ f ) δ(b − φ(D))ν

This shows that max |db f |2 ≤ bν · max Fµχ ≤ C6 (C1 , ν, R)δ −1

BR (x0 )

BR (x0 )

bν . (b − φ(D))ν

Theorem 4.4. Let (M 2m+1 , θ) be a noncompact complete pseudo-Hermitian manifold and (N, h) be a Riemannian manifold with sectional curvature K N ≤ κ for some κ ≥ 0. On B2R (x0 ) ⊂ M, R∗ ≥ −2mk,

|A|, |divA| ≤ k1

(4.21)

Assume that f : B2R (x0 ) ⊂ M → BD (p0 ) ⊂ N is pseudo-harmonic where BD (p0 ) is a regular ball in N. Then on BR (x0 ), there exists δ, ν and b > φ(D) only depending on φ(D) which are given in Lemma 4.1 such that max |db f |2 ≤ C6 (C1 , ν, R)δ −1

BR (x0 )

18

bν . (b − φ(D))ν

(4.22)

Lemma 2.3 says that if k = 0 and k1 = 0, then C1 = 0. According to (4.20), we find that   Cν 64ν 2 C6 (C1 , ν, R) = 2 + m(2 − ν)2 R

which implies that

2

max |db f | ≤

BR (x0 )



64ν 2 2+ m(2 − ν)2



Cν −1 bν δ → 0, as R → ∞. R (b − φ(D))ν

Hence we have the following Liouville theorem of pseudo-harmonic maps.

Theorem 4.5. Let (M, θ) be a noncompact complete Sasakian manifold with nonnegative pseudo-Hermitian Ricci curvature and (N, h) be a Riemannian manifold with sectional curvature K N ≤ κ for some κ ≥ 0. Then there is no nontrivial pseudo-Hermitian map from M to any regular ball of N.

5

Global Existence Theorem

Jost and Xu [18] studied the minimizing sequence of Dirichlet problem of sub-elliptic harmonic maps and obtained the existence theorem under some convexity condition. Their results [18] seem to depend on the global fields which satisfy the H¨ormander condition and the noncharacteristic assumption of the boundary. But the weak existence of Dirichlet problem and the interior continuity of weak solutions can be generalized to any sub-Riemannian manifolds with smooth boundaries, including pseudo-Hermitian manifolds. We formulate the results as follows: Theorem 5.1 (Theorem 1 and Theorem 2 in [18]). Suppose that (M, θ) is a pseudoHermitian manifold with smooth boundary and (N, h) is a Riemannian manifold with sectional curvature K N ≤ κ for some κ ≥ 0. Let BD = BD (p0 ) ⊂ N lie inside the cut locus of p0 ∈ M and D < 2√π κ . If ϕ ∈ S12 (M, N) satisfies ϕ(M ) ⊂ BD (p0 ), then there exists a weak pseudo-harmonic map f ∈ C(M, N) ∩ S12 (M, N) with and

2 f − ϕ ∈ S1,0 (M, N)

f (M ) ⊂ BD (p0 ).

Remark 5.2. Note that BD (p0 ) can be covered by the normal coordinate {z i } and thus it can be viewed as an open set of Rn where n = dim N. Hence the notion S12 (M, N) = S12 (M, Rn ), 2 and S1,0 (M, N) means the completion of all smooth Rn -valued functions with compact support under S12 -norm. Moreover, the weak pseudo-harmonic map f ∈ S12 (M, N) means that it satisfies X ∆b f i + Γijk (f )h∇b f j , ∇b f k i = 0 (5.1) j,k

in the distribution sense for all i = 1, 2, . . . n where f i = z i ◦ f and Γijk ’s are Christoffel symbols of Levi-Civita connection in (N, h). 19

Since the Euler-Lagrange equations of pseudo-harmonic maps are quasilinear sub-elliptic systems, these weak solutions will be smooth inside due to [31] by Xu-Zuily. Theorem 5.3 (Theorem 1.1 in [31]). Suppose that (M, θ) is a compact pseudo-Hermitian manifold (with or without boundary) and (N, h) is a Riemannian manifold. Let f : M → N be a weak pseudo-harmonic map and f ∈ S12 (M, N). If f is continuous inside M, then f ∈ C ∞ (M, N). Now we consider the global existence of pseudo-harmonic maps to regular balls. Suppose that (M, θ) is a complete noncompact pseudo-Hermitian manifold and (N, h) is a Riemannian manifold with sectional curvature K N ≤ κ for some κ ≥ 0. Let BD (p0 ) ⊂ N be a geodesic ball lying in the cut locus of p0 and D < 2√π κ . Assume that ϕ : M → BD (p0 ) with ϕ(x0 ) = p0 . We can choose an smooth exhaustion {Ωi } of M such that Ωi ⊃ B2i (x0 ). Theorem 5.1 and Theorem 5.3 guarantee that there is a smooth pseudo-harmonic map fi : Ωi → BD (x0 ). One can find the constants k(i) and k1 (i) such that R∗ B ≥ −2mk(i), and A|B2i , div A|B2i ≤ k1 (i). 2i

Hence fixed i, for j ≥ i, Theorem 4.4 controls the interior horizontal gradient of fj on Bi : max |dbfj |2 ≤ C7 (k(i), k1 (i), D, κ, i) Bi

where C7 (D, k(i), k1 (i), κ, i) only depends on k(i), k1 (i), D, κ, i. Arzel`a-Ascoli theorem yields that by taking subsequence, fj will uniformly converge to some continuous map in Bi as j → ∞. By diagonalization, some subsequence of {fi } will internally closed uniformly converge to a continuous map f : M → BD (p0 ) as i → ∞. Moreover, f is a weak solution of (5.1) and thus is smooth pseudo-harmonic by Theorem 5.3. Theorem 5.4. Suppose that (M, θ) is a complete noncompact pseudo-Hermitian manifold and (N, h) is a Riemannian manifold with sectional curvature K N ≤ κ for some κ ≥ 0. Let BD (p0 ) ⊂ N be a geodesic ball lying in the cut locus of p0 and D < 2√π κ . Then there is a pseudo-harmonic map f : M → BD (p0 ). It is notable that the pseudo-harmonic map given by Theorem 5.4 is always trivial if the domain has nonnegative pseudo-Hermitian Ricci curvature (cf. Theorem 4.5). At the end of this paper, we will give a nontrivial example when the domain has negative pseudo-Hermitian Ricci curvature. One model of Sasakian space form with constant negative pseudo-Hermitian sectional curvature is the Riemannian submersion π : BCn × R → BCn where BCn ⊂ Cn is the complex ball with Bergman metric Ω (cf. Example 7.3.22 in [4]). Let Ω0 be the canonical K¨ahler form on Cn . Since the identity I of BCn is a holomorphic map from BCn to Cn , then it is also a harmonic map from (BCn , Ω) to (Cn , Ω0 ). The lift of I is denoted by I˜ such that I˜ = I ◦ π : BCn × R → Cn . 20

ˆ is the Levi-Civita Suppose that ∇ is the Tanaka-Webster connection of BCn × R and ∇ connection associated with the Webster metric. Their relation is given by (cf. Lemma 1.3 in [13]) ˆ = ∇ − dθ ⊗ ξ + 2θ ⊙ J ∇

(5.2)

where 2θ ⊙ J = θ ⊗ J + J ⊗ θ. Then we have ˆ I˜ = ∇dI(dπ, ˆ ˆ ∇d dπ) + dI(∇dπ)

(5.3)

ˆ Assume where the Levi-Civita connections of (BCn , Ω) and (Cn , Ω0 ) are both denoted by ∇. 2n n that {ei }i=1 is a orthonormal frame in (BC , Ω) with ei+n = Jei for 1 ≤ i ≤ n and e˜i is the horizontal lift of ei . On one hand, the relation (5.2) guarantees that ˜ = Ξ(I)

2n X

˜ ei ) (∇e˜i dI)(˜

i=1

= = =

2n X

i=1 2n X

i=1 2n X i=1

2n   X ˜ ei ) − ˆ e˜ dI(˜ dI˜ (∇e˜i e˜i ) ∇ i i=1

2n   X   ˆ e˜ dI(˜ ˜ ei ) − ˜ ∇ ˆ e˜ e˜i ∇ d I i i i=1

ˆ e˜ dI)(˜ ˜ ei ). (∇ i

(5.4)

On the other hand, by the relation of Levi-Civita connection and metric, we have 2n X i=1



2n  X ˆ e ei ˆ ∇ dπ ∇e˜i e˜i = i i=1

which implies that 2n  X i=1

 ˆ ∇e˜i dπ (˜ ei ) = 0.

(5.5)

Taking the horizontal trace of (5.3) and using (5.4), (5.5), we obtain that ˜ = Ξ(I)

2n   X ˆ e dI (ei ) = 0, ∇ i i=1

since I is harmonic. Hence I˜ is nontrivial pseudo-harmonic. But the image of I˜ is exactly the unit ball in Cn which is a regular ball. So this is a nontrivial pseudo-harmonic example when the domain has negative pseudo-Hermitian Ricci curvature.

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Tian Chong School of Science, College of Arts and Sciences Shanghai Polytechnic University Shanghai, 201209, P. R. China [email protected] Yuxin Dong School of Mathematical Sciences Fudan University Shanghai, 200433, P. R. China [email protected] Yibin Ren College of Mathematics, Physics and Information Engineering Zhejiang Normal University Jinhua, 321004, Zhejiang, P.R. China [email protected] Wei Zhang School of Mathematics South China University of Technology Guangzhou, 510641, P.R. China [email protected]

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