An explicit formula for the Webster torsion of a pseudo-hermitian manifold and its application to torsion-free hypersurfaces

An explicit formula for the Webster torsion of a pseudo-hermitian manifold and its application to torsion-free hypersurfaces Song-Ying Li and Hing-Sun Luk Revised by June 16, 2006 To Sheng Gong, on his seventy-fifth birthday 1 Introduction Let M be a (2n + 1)-dimensional CR manifold with CR dimension n. Let H(M) be the holomorphic tangent bundle on M. We say that M is a strictly pseudoconvex pseudo-hermitian manifold in the sense of Webster [21] if there is a real one-form θ so that θ(x) 0 for all X H(M), and local complex one-forms {θ 1,, θ n } which form a basis for H (M) with (1.1) dθ i α,β1 h αβ θ α θ β and (h α β ) positive definite. (h αβ ) is determined by the Levi-form L θ with respect to θ. Then (1.2) dθ α θ γ ωγ α + θ τ α γ1 where τ α are in the linear span of θ 1,, θ n, and ω β α are 1-forms satisfying (1.3) 0 dh αβ h γβ ω γ α h αγ ω γ β dh αβ ω αβ ω βα. 1

In particular, if h αβ δ α β then (1.4) 0 dh αβ ω αβ ω βα ω β α ω α β, or ωβ α ω α β. Let (1.5) τ α h αγ τ γ A αβ θ β β1 Then the torsion on M is defined (see (2.14) below) as (1.6) Tor(u, v)(z) i(a αβ u α v β A αβ u α v β ) where u n+1 u j z j, v n+1 v j z j H z (M) and z M. Analysis and geometry on a strictly pseudoconvex pseudo-hermitian manifolds have been studied by many authors (see, for examples, [1], [2], [3], [4], [5], [7], [8, 9, 10], [11, 12], [13], [15], [17], [19], [20, 21] and references therein). From geometric point of view, it is important to understand curvatures and torsion. In [16], the authors give an explicit formula for the Webster pseudo Ricci curvature for hypersurfaces and provide an application for the characterization of the ball using the boundary pseudo scalar curvature. On many occasions, it is important to find an easier way to compute the Webster torsion. Various formulations through local coordinates were given in [20], [11, 12]. The first purpose of this paper is to give an explicit formula for the torsion of a real hypersurface in C n+1, which can be computed globally in terms of the defining function ρ. In other words, we shall prove the following theorem. THEOREM 1.1 Let M be a C 4 strictly pseudoconvex hypersurface in C n+1. Let ρ be a defining function for M which is C 3 in a neighborhood of M. Consider the pseudo-hermitian structure defined by θ 1 ( ρ ρ) 2i on M. Then for any u n+1 u j z j and v n+1 v j z j H z (M) we have (1.7) Tor(u, v)(z) Moreover, (1.8) Tor(u, v) 2 n+1 J(ρ) Re n+1 det H(ρ) 2 Re u(ρ q ) v(ρ q )(z), z M. J(ρ) u p v q( N det H(ρ) ) (ρ pq ). 2

where (1.9) N det H(ρ) ρ l l1, z l [ ρ and J(ρ) det ( ρ) ] ρ. H(ρ) Note: In the definition of ρ p appeared in (1.7), we require that H(ρ) is invertiable at z. However, the operator N in (1.8) is well-defined whether H(ρ) [ 2 ρ z j z k ] is invertiable on M or not since [det H(ρ)ρ pq ] is well defined as the transpose of the adjoint matrix of H(ρ). Moreover, N(ρ) J(ρ). (Some computations can be found in [13, 14]). In [11], Lee gave the general formula for the transformation of the torsion under a conformal change of the contact form in terms of covariant derivatives of the conformal factor. We will compute the covariant derivatives in the hypersurface case, isolating the role of the defining function, and give a transformation formula for the torsion under a conformal change. The second purpose of this paper is to apply our formula and obtain the explicit identification of the conformal class of the torsion-free hypersurface ( B n+1, θ), where B n+1 is the unit ball in C n+1. Let (1.10) θ 0 1 n+1 2i ( z j dz j z j dz j ) be the standard pseudo-hermitian structure for B n+1, which is torsion-free. Then we will prove the following theorem. THEOREM 1.2 Let h C 3 ( B n+1 ) be positive. Then ( B n+1, hθ 0 ) is torsion-free if and only if the harmonic extension of 1/h is a quadratic polynomial. Examples of torsion-free (normal or Sasakian) pseudo-hermitian manifolds were given by N. Tanaka in [19]; some sufficient conditions in terms of G homogeneous structures were given in Musso [18]. We will provide some more examples here. A real ellipsoid (1.11) E(a, b) {z C n+1 : ρ(z) 3 a j x 2 j + b j y 2 j 1 0}, a j, b j > 0

is a very typical example of strictly pseudoconvex hypersurfaces which was studied by S. Webster [21], who proves that (E(a, b), θ) with θ ( ρ ρ)/(2i) has vanishing Chern-Moser pseudo conformal curvature if and only if E(a, b) is complex linearly equivalent to the boundary of unit ball B n+1. The third purpose of this paper is to show that torsion-free real ellipsoid as above must be CR equivalent to the unit sphere of the same dimension. In particular, we will prove the following theorem. THEOREM 1.3 Any real ellipsoid (E(a, b), θ) in C n+1 is torsion-free if and only if it is the unit sphere after a complex linear change of variables. The paper is organized as follows. In section 2, we will prove Theorem 1.1. The transformation formula for the torsion under a conformal change will be given in Section 3. In Section 4, we will provide some examples of torsion-free or non-torsion-free real hypersurfaces. The proofs of Theorems 1.2 and 1.3 will be given in section 5. 2 Proof of Theorem 1.1 Let u be a real C 2 function on an open neighborhood D of M in C n+1 and H(u) [ 2 u z i z j ] (n+1) (n+1) be the complex hessian matrix of u. If H(u) is invertible then we let [u kj ] be the inverse of H(u) t so that (2.1) u kj u pj k1 u lk u lp δ kp. Let u k u z k, u j u z j and (2.2) u 2 u kj u k u j, u j k1 u jk u k and u j k1 u kj u k. Using this notation for ρ (a defining function for M), we write on M (2.3) θ i ρ i ρ, dθ i k,l1 ρ kl θ k θ l. 4

where (2.4) θ j dz j ih j θ, h j ρ j / ρ 2. We have (2.5) ρ j θ j 0. Let M 1 {z M : (z) 0}. It was proved in [21] as well as in [16] that on M 1, (2.6) dθ i where α,β1 h αβ θ α θ β (2.7) h αβ ρ αβ β ρ α ρ αn+1 ρ β + n+1 ρ α ρ β 2. Let (2.8) Y α z α ρ α Then for any f C 1 (M 1 ) we have n+1 z, Y i (h j n+1 j h j j ). (2.9) df f j dz j + f j dz j (Y α f θ α + Y α f θ α ) + Y (f)θ. α1 Notice that since θ α dz α ih α θ, we have (2.10) dθ α idh α θ ih α dθ θ γ ωγ α + θ τ α γ1 It was proved in [21] and in (2.16) (2.21) of [16] with the current notation, that ω α β can be chosen uniquely so that (2.11) dh αβ h γβ ω γ α h αγ ω γ β 0 5

and Then τ α i Y β h α θ β. β1 (2.12) τ α h αγ τ γ i h αγ Y β h γ θ β A αβ θ β β1 γ1 β1 with (2.13) A αβ ( i) h αγ Y β h γ. γ1 The torsion of M with respect to θ is defined as follows: (2.14) Tor θ (z α Y α, w β Y β ) i(a αβ z α w β A αβ z α w β ). For any u n+1 u j z j H z (M), we have n+1 (2.15) u u α Y α + u j ρ j n+1 u α Y α. α1 α1 Thus for any u, v H z (M), we have (2.16) Tor θ (u, v)(z) i α,β1 (A αβ u α v β A αβ u α v β ). The main point of this section is to give an explicit formula for computing the torsion; in other words, to give a proof of Theorem 1.1. It was proved in Section 2 in [16] that (2.17) h βγ h γ Y β log. γ1 Then (2.18) A αβ 6

i h αγ Y β (h γ ) γ1 i h αγ Y β ( ργ γ1 ρ ) 2 Y β (ρ γ ) i h αγ + i Y β( ρ 2 ) h γ1 ρ 2 ρ 2 αγ h γ γ1 i h ρ 2 αγ (ρ k Y β (ρ kγ ) + ρ kγ Y β (ρ k )) iy β (log ρ 2 )Y α log. γ1 Since h αγ ρ kγ γ1 (ρ αγ ρ αγ γ1 ρ αn+1 ρ γ + n+1 2 ρ αρ γ )ρ kγ δ αk ρ αn+1 ρ kn+1 ρ α δ kn+1 + ρ αn+1 ρ αn+1 ρ γ ρ kγ + n+1 δ αk ρ αn+1 ρ k ρ αρ 2 γ ρ kγ ρ α δ kn+1 + ρ αn+1 ρ k 2 δ αk ρ k Y α (log ) ρ α δ kn+1, ρ kn+1 (2.19) h αγ ρ kγ Y β (ρ k ) γ1 k1 (δ αk ρ k Y α (log ) ρ α δ kn+1 )Y β (ρ k ) Y β (ρ α ) Y α (log ) k1 ρ k Y β (ρ k ) ρ α Y β (log ). Notice that (2.20) Y β (ρ kγ ) ρ kq ρ pγ Y β (ρ pq ), 7

hence (2.21) Since (2.22) h αγ ρ k Y β (ρ kγ ) k1 γ1 n ( ρ k h αγ ρ kq ρ pγ Y β (ρ pq ) ) k1 γ1 p,q,k1 ρ k ρ kq Y β (ρ pq ) h αγ ρ pγ γ1 ρ q Y β (ρ pq )(δ αp ρ p Y α (log ) ρ q Y β (ρ αq ) + ρ p ρ q Y β (ρ pq ) k1 ρ p ρ q Y β (ρ pq ) ρ α δ pn+1 ) ρ p ρ q Y β (ρ pq )Y α log + ρ k Y β (ρ k ) + Y β ( ρ 2 ) ρ q Y β (ρ q ) + Y β ( ρ 2 ) ρ q Y β (ρ q ) ρ q (ρ βq ρ β ρ β 0, ρ β q ) ρ p Y β (ρ pq ρ q ) + Y β ( ρ 2 ) combining (2.18), (2.19), (2.21) and (2.22), we have ρ q Y β (q ) ρ α. A αβ i [ ρ q Y ρ 2 β (ρ αq ) ρ p ρ q Y β (ρ pq )Y α log ρ q Y β (q ) ρ α ] 8

i [ Yβ (ρ ρ 2 α ) Y α (log ) iy β (log ρ 2 )Y α (log ) i [ ρ q Y ρ 2 β (ρ αq ) i ( ρ p ρ q Y ρ 2 β (ρ pq ) k1 ρ k Y β (ρ k ) ρ α Y β (log ) ] ρ q Y β (q ) ρ α Y β (ρ α ) + k1 i [ Yβ (ρ ρ 2 α ) ρ αq Y β (ρ q ) Y β ( ) ρ α + i [ ρ ρ 2 αq Y β (ρ q ) + i Y ρ 2 α (ρ q )Y β (ρ q ) q Y β (ρ q ) ρ α Y β (ρ α ) + ρ α ρ α Y β ( ) ] ρ k Y β (ρ k ) + Y β ( ρ 2 ) ) Y α log ρ q Y β (ρ q ) ] z n+1 ρ α Y β ( ) ] Notice that for any w n+1 w j z j H z (M), we have w n α1 w α Y α. Therefore, for any u, v H z (M), we have (2.23) Tor θ (u, v) 2Re [ i Notice that on M α,β1 (2.24) J(ρ) ρ 2 det H(ρ). Hence (2.23) implies that (1.7) holds. To prove (1.8), we notice that A αβ u α v β] 2 n+1 ρ Re ( u(ρ 2 q )v(ρ q )). (2.25) and (2.26) u(ρ q )v(ρ q ) u(ρ q )ρ pq p1 u j δ pj u p. u(ρ q )v(ρ pq ρ p ) 9

p1 p1 u(ρ q )ρ pq v(ρ p ) + u p v(ρ p ) ρ pq u p v q ρ pq u p v q ρ pq u p v q p1 k,l1 k,l,p1 1 det H(ρ) p1 u(ρ q )ρ p u k ρ l v(ρ kl ) u k ρ l v p ρ pkl k,p1 u(ρ q )ρ p v(ρ pq ) k,l1 ρ kq ρ pl v(ρ kl ) u k v p N(ρ pk ) where N is defined by (1.9). Therefore, (2.27) Tor θ (u, v) 2 n+1 J(ρ) Re u p v q( N det H(ρ) ) (ρ pq ). This proves (1.8). Therefore, the proof of Theorem 1.1 is complete. 3 Torsion under conformal change of θ The purpose of this section is to consider the transformation formula of the torsion of a hypersurface when its pseudo-hermitian structure is changed in terms of the defining function. A general transformation formula for torsion in terms of covariant derivatives was obtained by J. Lee in [11]. Here we let r(z) h(z)ρ(z) with h(z) > 0, and consider (3.1) θ ρ 1 2i ( ρ ρ), θ r 1 ( r r). 2i Then θ r hθ ρ. In order to derive a formula for the torsion transformation, we first try to understand the relation between N r and N ρ. Lemma 3.1 If r(z) h(z)ρ(z) with J(r) > 0, then (3.2) T ρ,h h n 1 N r N ρ is a tangent vector field in H(M). 10

Proof. (3.3) Let [B ij (r)] be the co-factor matrix of H(r) so that B ij (r)r pj det H(r) δ ip. Since, on M, we have (3.4) r pq ρ p h q + ρ q h p + hρ pq, (3.5) N r B pq (r)r p h z q B pq (r)ρ p. z q For z 0 M, by a complex rotation, we may assume that z n+1 is complex normal at z 0. Thus ρ α (z 0 ) 0 for all 1 α n. Moreover, at z z 0 we have (3.6) B n+1n+1 (r) h n B n+1n+1 (ρ) and n (3.7) B n+1β (r)(z 0 ) h n B n+1β (ρ) h n 1 h α ( 1) α+β det H n (ρ) αβ. Thus T ρ,h h n 1 N r N ρ 2 h 1 α1 n α,β1 ( 1) α+β det H n (ρ) αβ h α z β is a tangent vector in H z0 (M). Therefore, the proof of the lemma is complete. For convenience, we introduce the following notation: (3.8) TOR ρ (u, v) 2 J(ρ) We will prove the following theorem. u p v q (N ρ det H(ρ))ρ pq (z). 11

THEOREM 3.2 If ρ is a defining function for M and r hρ with h > 0 and J(r) > 0, then for any u, v H z (M), (3.9) TOR r (u, v)(z) TOR ρ (u, v)(z) 2h + 2 J(ρ) u p v q 2 ( 1 ) z p z q h u p v q ( N ρ log h + T ρ,h ) ρpq Note: The relation between Tor θr (u, v)(z) Re TOR r (u, v)(z) and Tor θρ (u, v) follows by taking real parts on both sides of (3.9). Proof. Notice that (3.10) N r (r) J(r) h n+2 J(ρ) h n+2 N ρ (ρ). Also and hn r (ρ p ) N r (hρ p ) ρ p N r (h) N r r p N r (ρh p ) ρ p N r (h) (det H(r) N r(h) h )r p h p N r (ρ) N r r pq N r (h pq ρ + h p ρ q + h q ρ p + hρ pq ) h pq h N rr + N r (h p )ρ q + h p N r (ρ q ) + N r (h q )ρ p + N r (ρ p )h q + N r (h)ρ pq + hn r (ρ pq ) Thus for any u, v H z (M) we have u p v q (N r det H(r))r pq (z) u p v q ( det H(r)r pq + N r r pq ) 12

h h u p v q ( det H(r)hρ pq + N r (r) h pq h + h pn r (ρ q ) + h q N r (ρ p ) + N r (h)ρ pq + hn r (ρ pq )) +h n N ρ (ρ) u p v q ( ( det H(r) + N r (log h))ρ pq + N r ρ pq 2h n 1 N ρ (ρ)h p h q + h n N ρ (ρ)h pq ) u p v q ( det H(r) + N r (log h) + N r ) ρpq u p v q( 2h p h q + hh pq ) At z 0, by a complex rotation, we may assume that ρ α (z 0 ) 0 for all 1 α n. Then det H(r) N r (log h)(z 0 ) β1 β1 r n+1β B n+1β B n+1β r n+1 β log h B n+1β (hβ + h β + h n+1 ρ β h β ) B n+1β (r)(hβ + ρ β h n+1 ) β1 hb n+1n+1 (r)n+1 + h B n+1β (r)β + h n B n+1n+1 (ρ) h n+1 β1 n h n+1 det H(ρ) h n ( 1) α+β det H n (ρ) αβ h α β + h n B n+1n+1 (ρ) h n+1 α,β1 h n+1 det H(ρ) + h n n B αn+1 (ρ)h α + h n B n+1n+1 (ρ) h n+1 α1 n+1 h n+1 det H(ρ) + h n ρ β B αβ (ρ) h z α α,β1 h n+1 det H(ρ) + h n N ρ h Therefore, TOR r (u, v)(z) 13

2hn+2 J(r) +2 hn J(ρ) J(r) 2 J(ρ) + 2 J(ρ) u p v q ( det H(ρ)ρ pq h 1 N ρ hρ pq + N ρ ρ pq (N ρ h n 1 N r )ρ pq ) TOR ρ (u, v)(z) + 2 J(ρ) u p v q( 2h p h q + hh pq ) u p v q (N ρ det H(ρ))ρ pq (z) u p v q ( N ρ log h + T ρ,h ) ρpq 2h u p v q ( T ρ,h N ρ log h ) ρ pq 2h Therefore, the proof of the theorem is complete. u p v q 2 ( 1 ) z p z q h u p v q ( 1 ) pq. h 4 Examples of torsion-free real hypersurfaces in C n+1 In this section, we want to understand what kind of pseudo-hermitian manifolds (M, θ) of hypersurface type with zero torsion. We first point out a simple perturbation of the unit sphere in C n+1 which is not torsion-free. By Theoreon 1.1, Tor(u, v)(z) 0 for all u, v H z (M) and z M if and only if (4.1) Re u p v q( N det H(ρ) ) (ρ pq ) 0 for all u, v H z (M), z M. If we replace u by iu then (4.1) implies that (4.2) Re i Therefore, we have u p v q( N det H(ρ) ) (ρ pq ) 0 for all u, v H z (M), z M. (4.3) u p v q( N det H(ρ) ) (ρ pq ) 0 for all u, v H z (M), z M. 14

Thus (4.5) Tor θ 0 TOR ρ 0, θ i ρ. By formula (1.8), it is obvious that (S 2n+1, θ 0 ) with θ 0 1 n+1 2i (z j dz j z j dz j ) is a torsion-free pseudo-hermitian manifold. However, if one considers n+1 r(z) z 2 + ɛ z j 4 1, (4.6) D {z C n+1 : r(z) < 0}, M D, θ r 1 ( r r), 2i then we have the following proposition. Proposition 4.1 With the notation above, ( D, θ r ) is not a torsion-free pseudohermitian manifold for any ɛ > 0. Proof. Since (4.7) r p z p (1 + 2ɛ z p 2 ), H(r)(z) Diag(1 + 4ɛ z 1 2,, 1 + 4ɛ z n+1 2 ) and (4.8) r q p1 r pq r p 1 + 2ɛ z q 2 (1 + 4ɛ z q 2 ) z q, n+1 N det H(r) 1 + 2ɛ z q 2 1 + 4ɛ z q z 2 q, z q we have (4.9) r pq Therefore, N det H(r) r pq 2ɛz 2 pδ pq 2ɛδ pq 1 + 2ɛ z p 2 1 + 4ɛ z p 2 z p2z p 2ɛz 2 pδ pq (1 2 1 + 2ɛ z p 2 1 + 4ɛ z p ) 2 2ɛ 1 + 4ɛ z p 2 z2 pδ pq. J(r) 4 det H(r) TOR r(u, v)(z) ɛ 15 p1 z 2 p u p v p 1 + 4ɛ z p 2

where u, v H z ( D). It is obvious that TOR θr (u, v) 0 when z a e i for any 1 i n+1. When z D and z ae i for any a C and 1 i n+1, without loss of generality, we may assume z 1 0 and z n+1 0. Then we let u 1 z n+1 1 + 2ɛ z 1 2, un+1 z 1 1 + 2ɛ z n+1 2 v 1 z n+1 1 + 2ɛ z 1 2, vn+1 z 1 1 + 2ɛ z n+1 2 ; and u j v j 0 for all 2 j n. It is easy to see that u, v H z ( D) and J(r) 4 det H(r) TOR r(u, v) ɛz 2 1z 2 n+1 0, [ 1 (1 + 4ɛ z 1 2 )(1 + 2ɛ z 1 2 ) 2 + 1 (1 + 4ɛ z n+1 2 )(1 + 2ɛ z n+1 2 ) 2 ] and the proof of the proposition is complete. Next we let f(z) be a holomorphic function with f(0) 0 and α be a positive integer. In [19], Tanaka shows that if f is a weighted homogeneous polynomial, then M 0 {z C n+1 : z 2 1, f(z) 0} ( B n+1 ) Z(f) is a torsion-free pseudo-hermitian manifold of CR-dimension (n 1). Now we consider M D(f, α) where D(f, α) {z C n+1 : ρ(z) z 2 + f(z) 2α < 1}, f(0) 0, and θ α,f 1 2i ( ( z 2 + f(z) 2α ) ( z 2 + f(z) 2α )). We try to understand for what kind of f(z) and α is ( D(f, α), θ α,f ) torsionfree. For this purpose, we calculate TOR in terms of f and α as follows. Proposition 4.2 With notation above, we have (4.10) 1 n+1 2 J(ρ)TOR(u, v) [(1 z j j )f(z) α ] i, u i v j ( ij f(z) α ) Proof. Since ρ i z i + α f(z) 2 α 1 f(z) i f(z), ρ ij δ ij + α 2 f(z) 2(α 1) i f j f(z) 16

and Then where Thus and Then Hence and ρ ij (z) α f(z) 2 α 1 f(z) ij f(z) + α(α 1) f(z) 2 α 2 f(z) 2 i f j f, H(ρ) I n+1 + α 2 f(z) 2 α 1 ( f) ( f). H(ρ) 1 I n+1 α2 f(z) 2 α 1 ( f) ( f), 1 + A n+1 N det H(ρ) ρ q q A A(f)(z) α 2 f(z) 2 α 1 f 2. ρ pq δ pq α2 f(z) 2 α 1 q f(z) p f(z) 1 + A n+1 Nf det H(ρ) ρ q q f(1 (1 ρ q q α2 f(z) 2 α 1 1 + A A 1 + A ) 1 det H(ρ) 1 + A p1 n+1 1 + A ( α det H(ρ) N)f 1 n+1 1 + A (1 α z j ρ p p f q f q z q q f + A α f) )f z j Nρ ij αn( f(z) 2 α 1 f) ij f + α(α 1)N( f(z) 2 α 2 f(z) 2 ) i f(z) j f(z) Thus α 2 f(z) 2 α 1 Nf ij f + α 2 (α 1) f(z) 2 α 2 fnf i f j f α 2 f(z) 2 α 1[ ij f + (α 1) f(z) 2 1 f i f j f]nf ρ ij 1 det H(ρ) Nρ ij α f(z) 2 α 1 (f +α(α 1) f(z) 2 α 2 f(f α det H(ρ) Nf) ijf α det H(ρ) Nf) if j f 17

Therefore J(ρ) TOR(u, v) 2 det H(ρ)α f(z) 2 α 1 (f αrf)( det H(ρ)f(z) α 1 (f αrf) [(1 z j j )f(z) α ] i, i, i, u i v j ( ij f + α 1 i f j f)) f u i v j ( ij f(z) α ) u i v j ( ij f(z) α ) Thus, the proof of the proposition is complete. Proposition 4.3 With the notation above, the following two statements hold: (i) If f α is linear, then D(f, α) is biholomorphic to B n+1 and ( D(f, α), θ α,f ) is torsion free; (ii) If ( D(f, α), θ α,f ) is torsion free, then either f is a linear holomorphic function or f satisfies 2 f α u p v q (z) 0, z p z q fo all z D, u, v H z ( D). Proof. We first prove Part (i). Since f(z) α is linear holomorphic function, the vanishing of torsion is clear from Proposition 4.2. Let us write and f(z) α c j z j D {z C n+1 n+1 : z 2 + c j z j 2 < 1}. To prove that D is biholomorphic to the unit ball B n+1 in C n+1, it suffices to prove that there are (4.11) ϕ j (z) 18 k1 c jk z k

so that (4.12) or Let j,k1 n+1 ϕ j 2 z 2 + c jk c jl z k z l k,l1 c j z j 2 (δ kl + c k c l )z k z l (4.13) B [c jk ] n+1 n+1, C t (c 1,, c n+1 ). Since I n+1 +C C is positive definite, there is a non-singular matrix B so that (4.14) B B I n+1 + C C. Then ϕ : D B n+1 is a biholomorphism, and Part (i) is proved. To prove Part (ii). Since TOR(u, v) 0. If (1 n+1 z j j )f α 0 in an open set on D, then it equals 0 in an open set in C n+1 by unique continuation of holomorphic functions. This implies that f α must be a linear holomorphic function and α 1. Otherwise, (4.15) i, u i v j ( ij f(z) α ) 0 for all u, v H z ( D) and z D. The proof of the proposition is complete. Note: It seems that (4.15) implies that f(z) α is a linear function with α 1, but we cannot prove it at this moment. 5 Real Ellipsoids in C n+1 The real ellipsoid E(a, b) D(a, b) where (5.1) D(a, b) {z : a j x 2 j + b j y 2 j < 1}, a j, b j > 0, 19

is a very important example in pseudo-hermitian geometry and Chern-Moser theory. Many interesting questions on it were studied by S. Webster [20, 21], Huang and Ji [6], and many others. It is easy to see that E(a, b) can be written as (5.2) E(A, B) D(A, B), D(A, B) {z : ρ(z) < 1}, where (5.3) ρ(z) A j z j 2 + B j (z 2 j + z 2 j), A j a j + b j 2, B j a j b j. 4 Thus (5.4) ρ pq A p δ pq, ρ pq 2B p δ pq, N ρ (ρ) det H(ρ) A 1 j ρ j 2 Since ρ pq are constant, by (3.8) and (3.9), and det H(ρ) TOR ρ (u, v) 2 J(ρ) TOR hρ (u, v) TOR ρ (u, v) 2 N ρ log h J(ρ) Therefore TOR hρ (u, v) 0 if and only if u p v q ρ pq u p v q ρ pq 2h (5.5) 2( det H(ρ) N ρ log h) B p u p v p J(ρ)h p1 u p v q 2 ( 1 ) z p z q h u p v q pq ( 1 h ) 0. Without loss of generality, we may assume that A j 1 for all 1 j n + 1, otherwise, we can make a complex linear change of variables z j A j z j for 1 j n + 1. Then B j < 1/2 and (5.6) det H(ρ) 1, J(ρ) ρ 2 N ρ (ρ). 20

Thus (5.7) TOR hρ (u, v) 0 2(1+N ρ log h) Then we get the following theorem: p1 B p u p v p +J(ρ)h u p v q pq ( 1 h ) 0. THEOREM 5.1 Let E(B) {z C n+1 : ρ(z) z 2 + n+1 B j (zj 2 + z 2 j) 1 0} with B j < 1/2 and let h(z) C 3 (E(B)) be positive on E(B) with an extension in a neighborhood of E(B) so that N ρ log h 1 on E(B). Then the following statements are equivalent (i)e(b) B n+1, the sphere in C n+1, and n+1 u p v q 2 z p z q (1/h) 0 for all u, v H z (E(B)) and z E(B); (ii) Tor θhρ 0 on E(B) and n+1 u p v q 2 z p z q (1/h) 0 for all u, v H z (E(B)) for two points z 0 and z 1 in E(B) so that zj 0 0 and zj 1 0 for some 1 j n + 1. In particular, (E(B), θ ρ ) is torsion-free if and only if E(B) B n+1. Proof. First, we prove (i) implies (ii). When E(B) B n+1, B j 0 for all 1 j n + 1. Moreover, n+1 u p v q pq (1/h) 0 for all u, v H z ( B n ) and all z B n+1 implies (5.7), and therefore, Tor θhρ (u, v)(z) 0 for all u, v H z ( B n+1 ) for all z B n. Thus (ii) holds. Next we prove (ii) implies (i). By (4.5), Tor θhρ 0 on E(B) if and only if TOR hρ 0 on E(B). By (5.7), TOR hρ 0 on E(B) implies that (5.8) 2(1 + N ρ log h) B p u p v p + J(ρ)h p1 u p v q pq ( 1 h ) 0 for all u, v H z (E(B)) and z E(B). Since n+1 u p v q pq (1/h) 0 for all u, v H z (E(B)) at two points z z 0 and z 1 E(B), and since N ρ log h 1 on E(B), one has that (5.9) p1 B p u p v p 0 for all u, v H z (E(B) with z z 0 and z z 1. 21

Note that (5.10) ρ p z p + 2B p z p. Since z 0 j 0, (5.9) with u v e j implies that B j 0. If z 1 k 0 then (5.9) implies B k 0. Since z 1 j 0, for any k with z 1 k 0, (5.9) with u v implies that B k ρ j (z 1 ) 2 + B j ρ k (z 1 ) 2 0, Thus B k 0 for all 1 k n + 1. Therefore, E(B) B n+1. Applying (5.8) again, we have n u p v q pq (1/h)(z) 0 for all u, v H z ( B n ) and all z B n+1. The proof of the theorem is complete. and As a direct consequence, we have proved Theorem 1.3. Observe that for all u, v H z (E(B)) where (z) 0, u p v q pq u p v q (Y p Y q + ρ pq n+1 ), Y p p ρ p n+1, N ρ J(ρ) n+1 N ρ N n+1 ρ(ρ) n+1 ρ q Y q, hence (5.7) can be written as TOR hρ (u, v) 0 if and only if 2(1 + ρ q Y q log h) p1 B p u p v p + J(ρ)h Similarly, for all u, v H z (E(B)) where ρ l (z) 0, (5.11) 2(1+ ρ q Y q (l) log h) B p u p v p +J(ρ)h with Y q (l) q ρq ρ l l. Let X pl z l z p p1 u p v q Y p Y q ( 1 h ) 0. u p v q Y p (l)y q (l)( 1 h ) 0, z p z l. Then on B n+1, since ρ p z p, if Y p (l)y q (l)h(z) 0 then X pl X ql H(z) 0. The following question is natural and very interesting: Question: Let H C 2 ( B n+1 ) satisfy the following equation: (5.12) X pl X ql H 0, z B n+1 for all p, q l and 1 l n + 1. Then what can one say about H? 22

THEOREM 5.2 Let H C 2 (B n+1 ) be real, harmonic in B n+1 so that X pl X ql H(z) 0 on B n+1 for all 1 p, q, l n + 1. Then H is a quadratic polynomial. Proof. Since H is harmonic in B n+1, we claim that X pl H is also harmonic in B n+1. In fact, k1 2 1 X pl H(z) X pl z k z k n+1 4 H(z) 2 H 2 H (δ pk δ kl ) 0. z k z l z k z p Since X pl H is CR on B n+1, we have that X pl H can be extended to be holomorphic in B n+1. By the uniqueness of harmonic extension, we have that X pl H(z) is holomorphic in B n+1. Now let us write the Taylor series for H near z 0 as follow: (5.13) H(z) Then α + β 0 k1 a αβ z α z β. (5.14) X pl H(z) a αβ z α ( z p 2 β l z l 2 β p )z β ep e l α + β 1 α p0 α e p a α epβ+e l (β l + 1)z α z β + α e l a α el β+e p (β p + 1)z α z β a αβ+el (β l + 1)z α+ep z β α l 0 a αβ+ep (β p + 1)z α+e l z β (a α epβ+el (β l + 1) a α el β+e p (β p + 1))z α z β α e p+e l + a α epβ+el (β l + 1)z α z β + a αβ+el (β l + 1)z α+ep z β α e p,α l 0 α e l,α p0 α p0 a α el β+e p (β p + 1)z α z β α l 0 a αβ+ep (β p + 1)z α+e l z β (a α epβ+el (β l + 1) a α el β+e p (β p + 1))z α z β α e p+e l + a α epβ+el (β l + 1)z α z β + 2 a α epβ+el (β l + 1)z α z β α p>1,α l 0 23 α p1,α l 0

a α el β+e p (β p + 1)z α z β 2 a α el β+e p (β p + 1)z α z β α l >1,α p0 α l 1,α p0 + a α epβ+el (β l + 1)z α z β a α el β+e p (β p + 1)z α z β α p1,α l 1 α l 1,α p 1 (a α epβ+el (β l + 1) a α el β+e p (β p + 1))z α z β α p>1,α l >1 + (2a α epβ+el (β l + 1) 2a α el β+e p (β p + 1))z α z β α pα l 1 + (2a α epβ+el (β l + 1) a α el β+e p (β p + 1))z α z β α p1,α l >1 + (a α epβ+el (β l + 1) 2a α el β+e p (β p + 1))z α z β α p>1,α l 1 + a α epβ+el (β l + 1)z α z β + 2 a α epβ+el (β l + 1)z α z β α p>1,α l 0 α p1,α l 0 a α el β+e p (β p + 1)z α z β 2 a α el β+e p (β p + 1)z α z β α l >1,α p0 α l 1,α p0 Notice that X pl H(z) is holomorphic in B n+1. All terms b αβ z α z β in its power series expansion will vanish if β > 0. This fact with the above expansion imply that (i) If α l 0, α p 1 and β > 0 then a α ep β+e l 0. (ii) If α p 0, α l 1 and β > 0 then Thus, (i) and (ii) imply that a α el β+e p 0. (5.15) a αβ+el 0 if α l 0, β > 0; a αβ+ep 0 if α p 0, β > 0. (iii) If α l 1 and β > 0 then we have two cases: a) if α p > 1, then b) if α p 1, then a α epβ+e l (β l + 1) 2a α el β+e p (β p + 1); a α epβ+e l (β l + 1) a α el β+e p (β p + 1). 24

a) and b) imply that (5.16) a α+el β+e l (β l + 1) (2 δ 0αp )a α+epβ+e p (β p + 1) if α l 0. By (5.15) and (5.16), (5.17) a αβ+2el 0, if α l 1, β 0. (iv) If α p 1, α l > 1 and β > 0 then (5.18) 2a α epβ+e l (β l + 1) a α el β+e p (β p + 1); (v) If α p > 1, α l > 1 and β 1 then (5.19) a α epβ+e l (β l + 1) a α el β+e p (β p + 1) Consider α l 2. By (5.17) and (v), if α p > 1 then a α ep β+3e l β p + 1 β l + 3 a α e l β+2e l +e p 0, and by (5.17) and (iv), if α p 1, we have Thus a α epβ+3e l (β p + 1) 2(β l + 3) a α e l β+2e l +e p 0. (5.20) a αβ+3el 0 if α l 2. A similar computation for α l > 2, together with (5.15) and (5.17), gives (5.21) a αβ+(αl +1)e l 0 if α l 1, β 0; or α l 0, β 1. Since H is real-valued, we have Thus a βα a αβ (5.22) a αβ 0 if α β, α > 0, β 0; or α 0, β 2. 25

Therefore Let H(z) a 00 + (a α0 z α + a 0α z α ) + α 1 P k (z, z) α k α β 1 a αα z α 2 a αβ z α z β + α 2 a αα z α 2 where P k (z, z) is a homogeneous polynomial of degree 2k. Since X pl H(z) is holomorphic, one can see that X pl P k (z, z) X pl P k k2 k2 is holomorphic. Since X pl P k (z, z) is a homogeneous polynomial of degree 2k or 0, we have that X pl P k (z, z) is holomorphic for all k 2. We want to prove P k 0 when k 2. If k 2 then X pl P k (z, z) a αα z α (z p α l z α e l z l α p z α ep ) α k α k α k 1 α k 1 a αα z α+ep α l z α e l is holomorphic. Thus That is, α k a αα z α+e l α p z α ep a α+el α+e l z α+ep+e l (α l + 1)z α α k 1 [(α l + 1)a α+el α+e l (α p + 1)a α+ep α+e p ]z α+e l+e p z α a α+ep α+e p z α+e l+e p (α p + 1)z α (α l + 1)a α+el α+e l (α p + 1)a α+ep α+e p 0, if α > 0. (5.23) a α+el α+e l α p + 1 α l + 1 a α+e p α+e p, for all α > 0. 26

Since H is harmonic, we have It is easy to see that (5.24) α 1 0 H(z) 4 a αα + P k (z, z) α 1 k2 For convenience, we let a α a αα. Then a αα 0, P k (z, z) 0 for all k 2. P k (z, z) 4 Together with (5.23), we have α k 4 α k 4 a α a α α k 1 j z α 2 α 2 j z α e j 2 a α+ej (α j + 1) 2 z α 2 (5.25) a α+ej (α j + 1) 2 0, a α+el α p + 1 α l + 1 a α+e p for all α > 0. By (5.25), we have n+1 0 a α+e1 (α 1 + 1) 2 + (α j + 1)(α 1 + 1)a α+e1 (α 1 + 1)a α+e1 (α j + 1) j2 which implies that a α+e1 0, for all α 1. By the same argument, we have a α+ej 0, for all α 1, j 1, 2,, n + 1. Therefore (5.26) a α 0 for all α 2. 27

In other words, P k (z, z) 0 when k 2. Hence (5.27) H(z) b 0 + with b 0 being real, b jk b kj and 2Re (b j z j ) + j,k1 b jk z j z k (5.28) b jj 0. Thus H(z) is a harmonic quadratic polynomial. The proof of the theorem is complete. To prove Theorem 1.2. We let H(z) 1/h(z) be the harmonic extension of 1/h from B n+1 to B n+1. By (5.8) with B p 0 for 1 p n + 1, TOR hρ (u, v) 0 on B n+1 if and only if (5.29) X pl X ql (1/h) 0, z B n+1 for all 1 p, q, l n + 1. Since h is real-valued, by Theorem 5.2, we know that 1/h is a harmonic quadratic polynomial. Therefore, the proof of Theorem 1.2 is complete. References [1] S. M. Baouendi, P. Ebenfelt and L. P. Rothschild, Real Submanifolds in Complex Space and their Mappings, Math Series 47, Princeton University Press, Princeton, New Jersey, 1999. [2] R. Beals, P. Greiner and N. Stanton, The heat equation on a CR manifold. J. Differential Geom. 20 (1984), no. 2, 343 387. [3] D. C. Chang and S-Y. Li, A Riemann zeta function associated to sub- Laplacian on the unit sphere in C n, J. d Analyse Math., 86(2002), 86(2002), 25 48. [4] Najoua Gamara, The CR Yamabe conjecture the case n 1, J. Eur. Math. Soc., 3(2001), 105 137. 28

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[16] S.-Y. Li and H-S Luk, An explicit formula Webster pseudo Ricci curvature and its applications for characterizing balls in C n+1, Communication in Analysis and Geometry, to appear. [17] H-S. Luk, Affine connections and defining functions of real hypersurfaces in C n, Trans. Amer. Math. Soc. 259 (1980), no. 2, 579 588. [18] E. Musso, Homogeneous Pseudo-hermitian Riemannian Manifolds of Einstein Type, Amer. J. of Math, 113(1990), 219 241. [19] N. Tanaka, A Differential Geometric Study on Strongly Pseudo-Convex Manifolds, Kinokunija Company Ltd., Tokyo, 1975. [20] S. M. Webster, On the pseudo-conformal geometry of a Kähler manifold. Math. Z. 157 (1977), no. 3, 265 270. [21] S. M. Webster, Pseudohermitian geometry on a real hypersurface, J. Diff. Geom., 13 (1978), 25 41. Mailing Address: Partial work was done when the first author was visiting Fujian Normal University, China. The address is: School of Mathematics and Computer Secience, Fujian Normal University, Fuzhou, Fujian, China Current address for Li and Luk are: Department of Mathematics, University of California, Irvine, CA 92697 3875, USA. E-mail: sli@math.uci.edu Department of Mathematics, Lady Shaw Building, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong. E-mail: hsluk@math.cuhk.edu.hk 30