A NEW CHARACTERIZATION OF THE CR SPHERE AND THE SHARP EIGENVALUE ESTIMATE FOR THE KOHN LAPLACIAN. 1. Introduction

Transcription

1 A NEW CHARACTERIZATION OF THE CR SPHERE AND THE SHARP EIGENVALUE ESTIATE FOR THE KOHN LAPLACIAN SONG-YING LI, DUONG NGOC SON, AND XIAODONG WANG 1. Introduction Let, θ be a pseudoheritian anifold of diension + 1 and T the Reeb vector field. We always work with a local unitary frae {T α : α = 1,, } for T 1,0 and its dual frae {θ α }. Thus 1.1 dθ = 1 α θ α θ α. We will often denote T by T 0. In [LW], the first and third naed authors proved the following Obata type result in CR geoetry. Theore 1. Let be a closed pseudoheritian anifold of diension Suppose there is a real-valued nonzero function u C satisfying u α,β = 0, κ u α,β = + 1 u + 1 u 0 δ αβ, for soe constant κ > 0. Then is equivalent to the sphere S +1 with its standard pseudoheritian structure, up to a scaling. A weaker version of the above theore is also proved in diension 3 = 1 in [LW] which requires an additional condition. In this paper we prove a variant of the above theore which characterizes the CR sphere in ters of the existence of a non-trivial coplex-valued function satisfying a certain overdeterined syste. The precise stateent is the following theore. Theore. Let, θ be closed pseudoheritian anifold with diension Suppose that there exists a nonzero coplex-valued function f on satisfying f α,β = 0, f α,β = cfδ αβ for soe constant c > 0. Then, θ is CR equivalent to S +1 with its standard pseudoheritian structure, up to a scaling. Theore is otivated by the recent sharp lower bound for positive eigenvalues of the Kohn Laplacian by Chanillo, Chiu and Yang [CCY], just as Theore 1 is 1

2 SONG-YING LI, DUONG NGOC SON, AND XIAODONG WANG otivated by Greenleaf s sharp estiate for the first eigenvalue of the sublaplacian b. Recall that the Kohn Laplacian on a coplex-valued function f is defined by 1. b f = b b f = f α,α. and its conjugate b f = f α,α = b f 1T f. We have 1.3 b = b + b = b + 1T = b 1T. On a closed pseudoheritian anifold, the Kohn Laplacian b defines a nonnegative self-adjoint operator on the Hilbert space L of all coplex-valued functions f with f is integrable on, and the inner product on L is defined by 1.4 f 1, f = f 1 f. But unlike b, it does not satisfy the Hörander condition and, as a result, its resolvent is not copact. The three diensional case is ore coplicated than higher diensions. Nevertheless, it is proved by Burns and Epstein [BE] that the spectru of b in 0, consists of point eigenvalues of finite ultiplicity. In general, there ay exist a sequence of eigenvalues rapidly decreasing to zero. Zero is an isolated eigenvalue iff the range of b is closed. otivated by the ebedding proble for 3-diensional CR anifolds, Chanillo, Chiu and Yang [CCY] recently proved the following eigenvalue estiate for the Kohn Laplacian: Theore 3. Let be a closed 3-diensional pseudoheritian anifold. If the Paneitz operator P 0 is non-negative and the scalar curvature R κ, with κ being a positive constant, then any nonzero eigenvalue of b satisfies λ 1 κ. Recall that the Paneitz operator P 0 :C C on a closed pseudoheritian anifold is defined by 1.5 P 0 f = P α f,α = f γ,γαα + 1 A αβ f β. We say that P 0 is non-negative if for any f C 1.6 fp 0 f 0. Though Chanillo, Chiu and Yang only proved the eigenvalue estiate for 3-diensional pseudoheritian anifolds, their arguent can be easily generalized to higher diensions. In fact, since the Paneitz operator P 0 is always non-negative on closed pseudoheritian anifolds of diension 5, the stateent is even sipler see Chang and Wu [CW]. Theore 4. Let, θbe a closed pseudoheritian anifold of diension +1. Suppose for all X T 1,0 Ric X, X κ X, where κ is a positive constant. Then any nonzero eigenvalue of b satisfies λ + 1 κ.,α

3 A NEW CHARACTERIZATION OF THE CR SPHERE 3 Note that the estiate is sharp as equality holds on the sphere S +1 = { z C +1 : z = 1 } with the standard pseudoheritian structure θ 0 = 1 z S +1. A natural question is whether the equality case characterizes the CR sphere with the standard pseudoheritian structure up to a scaling. In their preprint [CW] Chang and Wu studied this proble and proved various partial results. One of the states that is indeed equivalent to the CR sphere S +1 if equality holds in Theore 4, provided that the following identity 1.7 A αβ f α f β = 0 is satisfied for a corresponding eigenfunction f. As a corollary of Theore, we can resolve this question in the general case. Naely, we have the following rigidity result. Corollary 1. If equality holds in Theore 4, then, θ is equivalent to the CR sphere S +1, θ 0, up to a scaling, i.e. there exists a CR diffeoorphis F : S +1 such that F θ 0 = cθ for soe constant c > 0. We expect that a siilar version of Theore is true in diension 3 fro which the characterization of the equality case in Theore 3 would follow. But we have not been able to prove it yet. This is due to an additional difficulty that arises only in 3-diensional case: It is not clear when functions annihilated by the CR Paneitz operator P 0 are CR-pluriharonic. Another reark is that despite the siilarity between these theores and Obata s theore in Rieannian geoetry, the proofs are essentially different due to the torsion of the Tanaka-Webster connection. In fact, a crucial step in the proof of Theore is to show that the torsion ust vanish. But unlike the approach in [LW], where the vanishing of torsion was deduced fro estiates regarding high powers of the real-valued function u or an eigenfunction of b, the vanishing of the torsion in our proof of Theore is proved by deriving various identities that are satisfied siultaneously only if the torsion is zero. The paper is organized as follows. In Section, we review soe basic facts in CR geoetry and discuss the Bochner forula for the Kohn Laplacian. In Section 3 we discuss the spectral theory of the Kohn Laplacian. In Section 4 we discuss the eigenvalue estiate of Chanillo, Chiu and Yang and its generalization to higher diensions. We deduce Corollary 1 fro Theore. The proof of Theore is then presented in Section 5. Acknowledgeent: We would like to thank Professor ei-chi Shaw for very instructive conversations on the analysis of the Kohn Laplacian on CR anifolds.. Preliinaries We first review soe basic facts in CR geoetry. Define the operator P : C A 1,0 by P f = f γ,γα + 1A αβ f β θ α.

4 4 SONG-YING LI, DUONG NGOC SON, AND XIAODONG WANG We write P α f = f γ,γα + 1A αβ f β. We also write B αβ f = f α,β 1 f γ,γδ αβ. The Paneitz operator P 0 :C C is defined by P 0 f = P α f,α = f γ,γαα + 1 A αβ f β Graha-Lee [GL] proved that P 0 is a real operator. oreover, P 0 is syetric, i.e. for f 1, f C with one of the copactly supported P 0 f 1 f = f 1 P 0 f. Here, the integrals are taken with respect to the volue for dv = θ dθ. They also proved the following identity when is closed:.1 B α,β f = B α,β f = 1 fp 0 f. Therefore, on a closed pseudoheritian anifold of diension + 1 5, the Paneitz operator is nonnegative, in the sense that for any coplex-valued function f, it holds that fp 0 f 0. But in diension 3, there are closed pseudoheritian anifolds whose Paneitz operator is NOT nonnegative. The following Bochner-type forula for the Kohn Laplacian was established by Chanillo, Chiu and Yang [CCY] see also Chang and Wu [CW]. Proposition 1. Let f be a coplex-valued function. Then b f = fα,β + fα,β + 1 bf α f α 1 f α b f α + R αβ f α f β 1 f αp α f + 1 f α Pα f. Integrating over a closed yields fα,β 0 = + f α,β + 1 b f α f α 1 f α b f α + R αβ f α f β 1 f αp α f + 1 f α Pα f fα,β = + B α,β f + 1 b f + 1 b f α f α 1 + R αβ f α f β + f α Pα f fα,β = + B α,β f + 1 b f + R αβ f α f β + f α Pα f,α. f α b f α

5 A NEW CHARACTERIZATION OF THE CR SPHERE 5 where we have used the decoposition f α,β = B α,β f + 1 bfδ αβ. Therefore, we have. + 1 b f = f α,β + Bα,β f +R αβ f α f β + f α Pα f. Integrating by parts yields f α Pα f = = = f P α f,α fp 0 f fp 0 f, In the last step, we have used the fact that P 0 is a real operator. Plugging this identity and.1 into. yields Proposition. Let be a closed pseudoheritian anifold of diension + 1 and f a coplex function on. Then b f = f α,β + R αβ f α f β + 1 fp 0 f. 3. The spectral theory of the Kohn Laplacian Based on the work of Beals and Greiner [BG], Burn and Epstein [BE] proved the following theore in diension 3. Theore 5. Let be a closed pseudoheritian anifold of diension 3. The spec b in 0, consists of point eigenvalues of finite ultiplicity. oreover all these eigenfunctions are sooth. The spectral theory of the Kohn Laplacian in the higher diensional case is in fact sipler. This is because the Hodge theory for 0, 1-fors is valid for all closed pseudoheritian anifold of diension by the fundaental work of Kohn [K]. The spectral theory for the Kohn Laplacian can then be deduced fro the Hodge decoposition theore for 0, 1-fors. This is known to the experts. But since it is not easily accessible in the literature, we give a detailed presentation, using the Bochner forula as a short cut. For background and a detailed exposition of the Kohn theory we refer to the book [CS] by Chen and Shaw, which is our priary source. Proposition 3. Let be a closed pseudoheritian anifold of diension +1 5 and f a coplex function on. Then 1 b f 1 = f 1 α,β + 1 f α,β + R σα f σ f α. Proof. When, we have by.1 1 fp 0 f = 1 1 = 1 1 B α,β f f α,β 1 bf. Plugging this identity into.3 yields the desired identity.

6 6 SONG-YING LI, DUONG NGOC SON, AND XIAODONG WANG Throughout the rest of this section, we assue. We have fro the previous Proposition 3.1 b f 1 f α,β + fα,β C b f, where C 0 depends on the pseudoheritian Ricci tensor. For f Do b we have b f = b f, f f b f. We will use this inequality iplicitly. Let H denote the space of L CR holoorphic functions, i.e. H = { f L : b f = 0 }. Proposition 4. The range of b : L L 0,1 is closed and ore precisely R b = b b Do. oreover, for all β R b, there exists a unique 0,1 b f H Do b such that b f = β and f C β. Proof. The first part is Corollary in [CS]. Suppose β = b u. By the Hodge decoposition for b on L 0,1 Theore in [CS] there exists α L 0,1 satisfying b b α = 0 and β = b bα. oreover α 1 C β. properties. It is easy to check that f := bα has all the desired Proposition 5. The range of b : L L is closed and ore precisely R b = H. oreover, for all φ H, there exists a unique f H Do b such that b f = φ and 3. f C b f. Proof. Clearly R b H. Suppose φ = b u R b. By Proposition 4, there is a unique f H such that b f = b u and f C b f. Then b f = b u = φ. We now prove that R b is closed. Suppose φ k = b f k φ in L 0,1, with each f k H. Then for k < l, we have b f k f l = φ k φ l. Thus, b f k b f l φk φ l f k f l C φ k φ l b f b f l. It follows that b f k b f l C φ k φ l. Applying 3. again yields f k f l C φ k φ l. Therefore, {f k } is Cauchy in L. Denote its liit by f. Then f k f and b f k φ. As b is a closed operator, we conclude that f Do b and b f = φ. Therefore, R b is closed. If R b was not the entire H, then there exists a nonzero φ H that is perpendicular to R b. This iplies φ H, obviously a contradiction.

7 A NEW CHARACTERIZATION OF THE CR SPHERE 7 Therefore, the operator b : H Do b H is bijective. The inverse operator exists and is denoted by T : H H. Naely, for each φ H, we define f = T φ H to be the unique solution to b f = φ which exists and unique by Proposition 5. Theore 6. The operator T : H H is copact. Proof. By 3.1 there exists a constant C > 0 such that for any f C f α,β + fα,β C b f + b f. It follows, by the Hörander estiate see Theore 8..5 in [CS], that 3.3 b f 1/ C b f + b f, where 1/ is the nor for the Sobolev space W 1/ 0,1. By approxiation, this inequality holds for all f Do b. We can further assue that f H. Suppose {f k } H Do b is a sequence such that φ k = b f k are bounded in L. By 3.3, b f k are bounded in W 1/ 0,1. By the Sobolev ebedding theore and passing to a subsequence, we can assue that { } b f k is Cauchy in L 0,1. By 3., {f k} is Cauchy in L. The proof is coplete. Theore 7. The spec b consists of countably any eigenvalues λ 0 = 0 < λ 1 < λ < with λ i as i. oreover, for i 1, each λ i is an eigenvalue of finite ultiplicity and all the corresponding eigenfunctions are sooth. Proof. We have proved, in Proposition, 5 that R b is closed. Thus, λ 0 = 0 is an eigenvalue whose corresponding eigenspace is H, which is of infinite diensional. With respect to the orthogonal decoposition L = H H, the operator λi b is given by the following atrix [ ] λi 0. 0 λi b H Therefore, λ > 0 is in spec b if and only if λ 1 is in spec T. As T is copact, spec T 0, + consists of countably any eigenvalues of finite ultiplicities µ 1 > µ > with li µ i = 0. Therefore, spec b consists of λ 0 = 0 < λ 1 < λ < with λ i = 1/µ i for all i 1 and the eigenspace of b corresponding to λ i equals the eigenspace of T corresponding to µ i. Suppose f is an eigenfunction corresponding to an eigenvalue λ > 0, i.e. b f = λf. Then the 0, 1-for β = b f satisfies b β = b b f = λ b f = λβ. By the Hodge theory for 0, 1-fors, β is sooth. As f = 1 λ bβ, we see that f is sooth.

8 8 SONG-YING LI, DUONG NGOC SON, AND XIAODONG WANG 4. The eigenvalue estiate With the spectral theory of b understood, we can now state the following Theore 8. Let be a closed pseudoheritian anifold of diension + 1. When = 1, we further assue that the Paneitz operator is non-negative. Suppose Ric X, X κ X, where κ is a positive constant. Then any nonzero eigenvalue of b satisfies λ + 1 κ. Proof. Suppose f is a nonzero eigenfunction with eigenvalue λ > 0. By.3, under the assuptions + 1 λ f = f α,β + R αβ f α f β + 1 fp 0 f κ f = κ f b f = λκ f. Thus, λ +1 κ. Theore 8 was first proved by Chanillo, Chiu and Yang [CCY] in the case = 1. We have followed basically the sae arguent see also Chang and Wu [CW]. +1κ in Theore 8 and f a corresponding eigen- Proposition 6. Suppose λ = function. Then we ust have: i If = 1, then ii If, then f 1,1 = 0, f 1,1 = κ f, P 0f = 0; f α,β = 0, f α,β = κ + 1 fδ αβ. Proof. If equality holds, by inspecting the proof of Theore 8, we ust have f α,β = 0 and fp 0f = 0. As P 0 is nonnegative, it follows easily that P 0 f = 0. As P 0 is a real operator, we also have P 0 f = 0. When, this iplies by.1 that f α,β = 1 bfδ αβ = κ + 1 fδ αβ. Cobining this Proposition and Theore, we iediately obtain the following Corollary. Suppose λ = +1κ in Theore 8 and. Then, θ is CR equivalent to S +1 with its standard pseudoheritian structure, up to scaling.

9 A NEW CHARACTERIZATION OF THE CR SPHERE 9 5. Proof of the ain Theore We now prove our ain theore Theore. c = 1/. Theore is equivalent to the following By scaling, we ay assue Theore 9. Let, θ be closed pseudoheritian anifold with diension Suppose that there exists a nonzero coplex function f on satisfying f α,β = 0, f α,β = 1 fδ αβ Then, θ is CR equivalent to the S +1 with its standard pseudoheritian structure Therefore we have f α,β = 0, f α,β = 1 fδ αβ. Using 5.1 and 5. it is easy to derive b 5.3 f = 1 α ff α. Proposition 7. We have A αβ f γ = A αγ f β, f γ = 1A γσ f σ. Proof. Differentiating 5.1 yields A αβ f γ = A αγ f β. Differentiating 5. yields δ αβ f γ / = f α,βγ = f α,γβ 1 δ αβ A γσ δ αγ A βσ f σ = δ αγ f β / 1 δ αβ A γσ δ αγ A βσ f σ. Hence δ αβ fγ / 1A γσ f σ = δαγ fβ / 1A βσ f σ. Therefore f γ / 1A γσ f σ = 0. Let Q = 1A αβ f α f β. Set K = { p : b f p = 0 }. On \K we define ψ = Q/ b f. Note that ψ is sooth and bounded on \K. Lea 1. \K is open and dense.

10 10 SONG-YING LI, DUONG NGOC SON, AND XIAODONG WANG Proof. We need to prove that K has no interior point. Write f = u + 1v with u and v real. Then, using 5. u α,β = 1 f α,β + f α,β = 1 f β,α 1f 0 δ αβ + f α,β = 1 f + 1f0 + f δ αβ. 4 Therefore, u is CR pluriharonic. Siilarly, v is also CR pluriharonic. Now suppose b f = 0 on a connected open set U. By 5. f = 0 on U. Hence, u and v both vanish on U. Being CR pluriharonic, u and v then ust be identically zero on. This is a contradiction. Proposition 8. On \K we have ψ A αβ = b f f αf β, f γ = ψf γ. Proof. Using 5.4, we copute A αβ b f = A αγ f β f γ = A γα b f f β f γ b f f β f γ = A γσ f α f σ b f 1Q = b f f αf β = 1 ψ f αf β. This proves 5.6. To prove 5.7, we copute using 5.5 and 5.6 f γ = 1A γσ f σ = ψ b f f γf σ f σ Reark 1. Fro 5.6, we obtain on \K = ψf γ. A := α,β A αβ = 1 4 ψ. Therefore, ψ extends soothly to the entire.

11 A NEW CHARACTERIZATION OF THE CR SPHERE 11 Proposition 9. On \K we have 5.8 b ψ = 0 and 5.9 1f0 1 f + 1 ψf = 0. Proof. Differentiating f α = ψf α and using 5. yields f α,β = ψ β f α + ψf α,β = ψ β f α 1 ψfδ αβ. By using 5. again, we further copute the left hand side Therefore, f α,β = f β,α + 1f 0 δ αβ ψ β f α = = 1 fδ αβ + 1f 0 δ αβ. 1f0 1 f + 1 ψf δ αβ. Fro this, it follows easily since that 1f 0 f/ + ψf/ = 0 and ψ β = 0. Proposition 10. We have 5.10 R αβ f α = + 1 f β. Proof. Using 5.1 and 5., we copute 0 = f α,βγ = f α,γβ + 1δ βγ f α,0 R βγασ f σ = 1 f βδ αγ + 1δ βγ f 0,α A ασ f σ Rβγασ f σ. Differentiating 5.9 and using 5.8 and 5.7 yields 1f 0,α = 1 ψfα f α = 1 ψ 1 f α. Plugging this into the previous equation and using 5.7 again, we obtain 0 = 1 [ 1 f βδ αγ R βγασ f σ + ψ 1 f α ] 1ψA ασ f σ δ βγ = 1 f βδ αγ R βγασ f σ 1 f αδ βγ, where in the last step, we have used 5.6. Therefore, R βγασ f σ = 1 f β δ αγ + f α δ βγ. Taking trace over β and γ yields 5.10.

12 1 SONG-YING LI, DUONG NGOC SON, AND XIAODONG WANG Since the Paneitz operator P 0 is real, we have fp 0 f = fp 0 f = 0. Applying the Bochner forula to f yields + 1 b f = f α,β + R αβ f α f β + 1 fp 0 f = f α,β ψ b f. We copute on \K, using 5.7 and Fro this, we get on \K f α,β = ψ β f α + ψf α,β = ψ β f α. 5.1 b ψ b f = α,β f αβ. Notice that the right hand side is sooth on. Therefore, b ψ b f extends soothly to the entire and the above inequality holds on. Siilarly, using 5.8 as well b f = f α,α = ψf α,α = ψf α,α = ψf. Plugging these into the integral identity, we obtain + 1 ψ f 4 = b ψ b f ψ b f, i.e b ψ b f = + 1 ψ f b f. Lea. We have on \K b ψ = 0. Proof. By 5.11, we have ψ α f β = ψ β f α. Therefore, on \K 5.14 ψ α = ψ β f β f b α f. β Fro this, we get b ψ b f = ψ β f β. β

13 A NEW CHARACTERIZATION OF THE CR SPHERE 13 Since b ψ = 0, we have ψ α,β = 1ψ 0 δ αβ. For each α we copute using 5.14 and Proposition 6 ψ αα = ψ β,α f β f α + ψ β f β f α,α b f β β β ψ β f β f α b f 4 γ f γ f γ,α = ψ α,α f α 1 β ψ β f β f b f + 1 β ψ β f β f f α b f 4. Hence, ψ αα 1 b f f α = 1 β ψ β f β f b f 1 b f f α. It follows that on \K ψ αα = 1 β ψ β f β b f Set B = β ψ β f β. Note that B is a sooth function on. Then on \K, as b ψ = ψψ αα = 1 ψψ β f β b f = 1 Bf b f. β We copute on \K, using Proposition 6 and 5.15 f ψ f α + b f ψ α,α = f ψ f α + b f ψψ α,α = fb + ψ b f ψ f fψf α ψ α + b f ψψ α,α + b ψ = 1 fb fb + ψ b f f + b f b ψ. Since both sides are sooth on and \K is open and dense, the above identity holds on the entire. Integrating over and taking the real part yields b f b ψ = ψ f b f. Cobining this identity with 5.13, we obtain b ψ = 0 on \K. Lea 3. ψ = 0 and therefore, the torsion vanishes. b f b ψ = 0. Therefore,

14 14 SONG-YING LI, DUONG NGOC SON, AND XIAODONG WANG Proof. By Proposition 9 and Proposition, b ψ = 0 and b ψ = 0 on \K. Therefore, ψ is locally constant on \K. Since ψ extends soothly to, ψ is constant on. Differentiating 5.6 on \K using Proposition 6 and 5.3, we get on \K Hence, A αβ,γ b f = A αβ ff γ. A αβ,γ f α f β f γ b f = A αβ f α f β f b f = 1Qf b f. Thus, on \K, we have Qf = 1A αβ,γ f α f β f γ, i.e. ψf = 4 1A αβ,γ f α f β f γ / b f. Fro this, we obtain first on \K and then, by continuity, on the whole as ψ is continuous on and \K is open and dense 5.16 ψ f 4C b f, where, C = ax Aαβ,γ. Let p 0 be a point where f achieves its axiu. Suppose b f p 0 0, i.e p 0 \K. Then differentiating at p 0 and using 5.7, we have Hence, 0 = ff α + f α f = f α f + ψf ψ p 0 = f p 0 f p 0. As ψ is constant, we have ψ 1. By 5.9 we have 1f 0 = 1 \K. Differentiating and using 5.7 yields 1f0α = 1 fa ψf α = 0, 1f0α = 1 fa ψf α f ψf on = 1 1 ψ f α = 0. Therefore, f 0 is constant. As f 0 = 0, we ust have f 0 0. Then 5.9 reduces to f = ψf. At p 0 this yields ψ p 0 = f p 0 /f p 0. This is contradictory to Therefore, b f p 0 = 0, then the inequality 5.16 iplies ψ p 0 = 0. Consequently, ψ 0.

15 A NEW CHARACTERIZATION OF THE CR SPHERE 15 Since ψ 0, we have, in view of and 5.9 A αβ = 0, b f = 0, 1f0 = 1 f. Write f = u + 1v in ters of its real part u and iaginary part v. Fro the above identities and Proposition 6 it is easy to prove the following Proposition 11. We have v = u 0, while u satisfies u αβ = 0, u α,β = u + u 0 δ αβ. With this proved, we can now apply the result of Li-Wang [LW] Theore 5 therein to conclude that is CR equivalent to S +1 with its standard pseudoheritian structure. In fact, we do not need that result in its full generality as we have the additional condition A = 0 at our disposal. Since the arguent there under the additional condition A = 0 is siple, we give an outline for copleteness. Let g θ be the adapted Rieannian etric and D u the Rieannian Hessian. Then fro the above identities we obtain by standard calculation see Proposition in [LW] D u = u 4 g θ, here we used the fact that the torsion A = 0. By Obata s theore [O],, g θ is isoetric to the sphere S +1 with the etric g 0 = 4g c, where g c is the canonical etric. Without loss of generality, we can take, g θ to be S +1, g 0. Then θ is a pseudoheritian structure on S +1 whose adapted etric is g 0 and the associated Tanaka-Webster connection is torsion-free. It is a well known fact that the Reeb vector field T is then a Killing vector field for g 0 see Reark 1 in [LW]. Therefore there exists a skew-syetric atrix A such that for all X S +1, T X = AX, here we use the obvious identification between z = z 1,..., z +1 C +1 and X = x 1, y 1,..., x +1, y +1 R +. Changing coordinates by an orthogonal transforation we can assue that A is of the following for A = where a i 0. Therefore 0 a 1 a 1 0 T = i a i Since T is of unit length we ust have 4 i 0 a+1 a +1 0 y i x i x i y i a i x i + y i = 1

16 16 SONG-YING LI, DUONG NGOC SON, AND XIAODONG WANG on S +1. Therefore all the a i s are equal to 1/. It follows that This finishes the proof of Theore. θ = g 0 T, = 1 z 1. References [BG] R. Beals and P. Greiner, Calculus on Heisenberg anifolds. Ann. of ath. Stud., vol. 119, Princeton Univ. Press, New Jersey, [BE] D. Burns; C. Epstein, Ebeddability for Three-diensional CR anifolds, J. Aer. ath. Soc , [CCY] S. Chanillo; H.-L. Chiu; P. Yang, Paul, Ebeddability for 3-diensional Cauchy-Rieann anifolds and CR Yaabe invariants. Duke ath. J , no. 15, [CW] S.-C. Chang; C.-T. Wu, On the CR Obata Theore for Kohn Laplacian in a Closed Pseudoheritian Hypersurface in C n+1. Preprint, 01. [CS] S.-C. Chen;.-C. Shaw, Partial differential equations in several coplex variables. AS/IP Studies in Advanced atheatics, 19. Aerican atheatical Society, Providence, RI; International Press, Boston, A [GL] C. Robin Graha; John. Lee, Sooth solutions of degenerate Laplacians on strictly pseudoconvex doains. Duke ath. J , no. 3, [K] J. J. Kohn, Boundaries of Coplex anifolds, Proc. Conf. Coplex anifolds inneapolis, 1964, Springer-Verlag, New York, 81-94, [LW] S.-Y. Li; X. Wang, An Obata-type Theore in CR Geoetry, arxiv: , to appear in J. Diff. Geo. [O]. Obata, Certain conditions for a Rieannian anifold to be isoetric with a sphere. J. ath. Soc. Japan Departent of atheatics, University of California, Irvine, CA 9697 E-ail address: sli@ath.uci.edu Departent of atheatics, University of California, Irvine, CA 9697 E-ail address: snduong@ath.uci.edu Departent of atheatics, ichigan State University, East Lansing, I 4884 E-ail address: xwang@ath.su.edu