AN OBATA-TYPE THEOREM IN CR GEOMETRY. 1. Introduction

Transcription

1 AN OBATA-TYPE THEORE IN CR GEOETRY SONG-YING LI AND XIAODONG WANG Abstract. We discuss a sharp lower bound for the first positive eigenvalue of the sublaplacian on a closed, strictly pseudoconvex pseudoheritian anifold of diension We prove that the equality holds iff the anifold is equivalent to the CR sphere up to a scaling. For this purpose, we establish an Obata-type theore in CR geoetry which characterizes the CR sphere in ters of a nonzero function satisfying a certain overdeterined syste. Siilar results are proved in diension 3 under an additional condition. 1. Introduction In Rieannian geoetry, estiates on the first positive eigenvalue of the Laplace operator have played iportant roles and there have been any beautiful results. We refer the reader to the books Chavel [C] and Schoen-Yau [SY]. The following theore is a classic result. Theore 1. Lichnerowicz-Obata Let n, g be a closed Rieannian anifold with Ric n 1 κ, where κ is a positive constant. Then the first positive eigenvalue of Laplacian satisfies 1.1 λ 1 nκ. oreover, equality holds iff is isoetric to a round sphere. The estiate λ 1 nκ was proved by Lichnerowicz [L] in The characterization of the equality case was established by Obata [O] in 196. In fact, he deduced it fro the following ore general Theore. Obata [O] Suppose N n, g is a coplete Rieannian anifold and u a sooth, nonzero function on N satisfying D u = c ug, then N is isoetric to a sphere S n c of radius 1/c in the Euclidean space R n+1. In CR geoetry, we have the ost basic exaple of a second order differential operator which is subelliptic, naely the sublaplacian b. On a closed pseudoheritian anifold, the sublaplacian b still defines a selfadjoint operator with a discrete spectru 1. λ 0 = 0 < λ 1 λ with li k λ k = +. One would naturally hope that the study of these eigenvalues in CR geoetry will be as fruitful as in Rieannian geoetry. An analogue of the Lichnerowicz estiate for the sublaplacian on a strictly pseudoconvex pseudo- Heritian anifold +1, θ was proved by Greenleaf in [G] for 3 and by Li and Luk [LL] for =. Later it was pointed out that there was an error in The second author was partially supported by NSF grant DS

2 SONG-YING LI AND XIAODONG WANG the proof of the Bochner forula in [G]. Due to this error, the Bochner forula as well as the CR-Lichnerowicz theore in [G, LL] are not correctly forulated. The corrected stateent is Theore 3. Let, θ be a closed, strictly pseudoconvex pseudoheritian anifold of diension Suppose that the Webster pseudo Ricci curvature and the pseudo torsion satisfy 1.3 Ric X, X + 1 T or X, X κ X for all X T 1,0, where κ is a positive constant. Then the first positive eigenvalue of b satisfies 1.4 λ κ. The estiate is sharp as one can verify that equality holds on the CR sphere S +1 = {z C +1 : z = 1} with the standard pseudoheritian structure θ 0 = 1 z 1. The natural question whether the equality case characterizes the CR sphere was not addressed in [G]. The torsion appearing in 1.3 is a ajor new obstacle coparing with the Rieanian case. This question has been recently studied by several authors and partial results have been established. Chang and Chiu [CC1] proved that the equality case characterizes the CR sphere if has zero torsion. In [CW], Chang and Wu proved the rigidity under the condition that the torsion satisfies certain identities involving its covariant derivatives. Ivanov and Vassilev [IV] proved the sae conclusion under the condition that the divergence of the torsion is zero. Li and Tran [LT] considered the special case that is a real ellipsoid EA in C +1. They coputed κ explicitly and proved that the equality, λ 1 = κ/ + 1 iplies EA is the sphere. But in general it is very difficult to handle the torsion. In this paper, we provide a new ethod which can handle the torsion and yields an affirative answer to this question in the general case. Theore 4. If equality holds in Theore 3, then, θ is equivalent to the sphere S +1 with the standard pseudoheritian structure 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. In fact our proof yields the following ore general result which can be viewed as the CR analogue of Theore for notation see Section. Theore 5. Let be a closed pseudoheritian anifold of diension Suppose there exists a real nonzero function u C satisfying u α,β = 0, κ u α,β = + 1 u + 1 u 0 h αβ, for soe constant κ > 0. Then is equivalent to the sphere S +1 with the standard pseudoheritian structure up to a scaling.

3 AN OBATA-TYPE THEORE IN CR GEOETRY 3 The 3-diensional case is ore subtle. It is not clear if these results are true in 3 diensions. Partial results with additional conditions are discussed in the last section. The approach in [CC1] is to consider a faily of adapted Rieannian etrics and try to apply the Lichnerowiz-Obata theore. This approach requires very coplicated calculations to relate the various CR quantities and the corresponding Rieannian ones. In Deceber 010, the authors found a new approach working directly with the Rieannian Hessian of the eigenfunction. With this approach we generalized the Chang-Chiu result to show that rigidity holds provided the double divergence of the torsion vanishes see Reark 5 in Section4. In their preprint [IV], Ivanov and Vassilev found the sae strategy independently and proved rigidity under the condition that the divergence of the torsion is zero. But to solve the general case requires a new ingredient. We eploy a delicate integration by parts arguent which requires a good understanding of the critical set of the eigenfunction. The paper is organized as follows. In Section, we review soe basic facts in CR geoetry. In Section 3, following the arguent of Greenleaf we present the proof of Theore with all the necessary corrections. Theore 3 is proved in Sections 4 and 5. Finally, in Section 6, we discuss the 3-diensional case. Acknowledgeent. The authors wish to thank the referees for carefully reading the paper and aking valuable suggestions.. Preliinaries Let, θ, J be a strictly pseudoconvex pseudoheritian anifold of diension +1. Thus G θ = dθ, J defines a Rieannian etric on the contact distribution H = ker θ. As usual, we set T 1,0 = {w 1Jw : w H } T C and T 0,1 = T 1,0. Let T be the Reeb vector field and extend J to an endoorphis φ on T by defining φ T = 0. We have a natural Rieannian etric g θ on such that T = RT H is an orthogonal decoposition and g θ T, T = 1. In the following, we will siply denote g θ by,. Let be the Levi-Civita connection of g θ while is the Tanaka-Webster connection. For basic facts on CR geoetry, one can consult the recent book [DT] or the original papers by Tanaka [T] and Webster [W]. Recall that the Tanaka-Webster connection is copatible with the etric g θ, but it has a non-trivial torsion. The torsion τ satisfies τ Z, W = 0, τ Z, W = ω Z, W T, τ T, J = Jτ T, for any Z, W T 1,0, where ω = dθ. We define A : T T by AX = τ T, X. It is custoary to siply call A the torsion of the CR anifold. It is easy to see that A is syetric. oreover AT = 0, AH H and AφX = φax. The following forula gives the difference between the two connections and. The proof is based on straightforward calculation and can be found in [DT].

4 4 SONG-YING LI AND XIAODONG WANG Proposition 1. We have X Y = X Y + θ Y AX + 1 θ Y φx + θ X φy [ AX, Y + 1 ] ω X, Y T. Reark 1. We have X T = AX + 1 φ X. In particular, T T = 0. If X and Y are both horizontal i.e. X, Y H, then X Y = X Y [ AX, Y + 1 ] ω X, Y T. In the following, we will always work with a local frae {T α : α = 1,, } for T 1,0. Then {T α, T α = T α, T 0 = T } is a local frae for T C. Let h αβ = iω T α, T β = g θ T α, T β be the coponents of the Levi for. For a sooth function u on, we will use notations such as u α,β to denoted its covariant derivatives with respect to the Tanaka-Webster connection. Let D u be the Hessian of u with respect to the Rieannian etric g θ. By straightforward calculation using Proposition 1, one can derive Proposition. We have the following forulas D u T, T = u 0,0, 1 D u T, T α = u α,0 u α, D u T α, T β = u α,β + A αβ u 0, 1 D u T α, T β = u α,β h αβ u 0. In doing calculations we will need to use repeatedly the following forulas which can be found in [DT] or [Lee]. Proposition 3. We have the following forulas u 0,α = u α,0 + A β αu β, u α,β = u β,α, u α,β = u β,α + 1h αβ u 0, u α,0β = u α,β0 + A γ β u α,γ + R σ β0αu σ = u α,β0 + A γ β u α,γ A αβ,γ h σγ u σ, u α,0β = u α,β0 + u a,γ h γν A νβ + h γν A νβ,α u γ, u α,βγ = u α,γβ + 1 h αβ A σ γ h αγ A σ u β σ, u α,βγ = u α,γβ + 1h βγ u α,0 R σ βγαu σ, u α,βγ = u α,γβ R σ βγαu σ = u α,γβ + 1 A αγ u β A αβ u γ.

5 AN OBATA-TYPE THEORE IN CR GEOETRY 5 Reark. Our convention for the curvature tensor is R X, Y, Z, W = X Y Z + Y X Z + [X,Y ] Z, W. 3. The estiate on λ 1 In this section, we prove the estiate on λ 1 following Greenleaf [G]. This serves two purposes. First, there is a istake in [G] as pointed out by [GL] and [CC1]. This has caused soe confusion e.g. see the presentation in [DT] and we hope to clarify the whole situation. Secondly, we need to analyze the proof when we address the equality case. Fro now on, we always work with a local unitary frae {T α : α = 1,, } for T 1,0. Given a sooth function u, its sublaplacian is given by b u = α We have the following Bochner forula. u α,α + u α,α. Theore 6. Let b u = u α u α. Then 1 b b u = u α,β 1 + u α,β + [ bu α u α + b u α u α ] +R ασ u σ u α + 1 [Aασ u α u σ A ασ u σ u α ] + 1 u β u β,0 u β u β,0. Reark 3. This was first derived by Greenleaf [G]. But due to a istake in calculation pointed out by [GL] and [CC1], the coefficient on the RHS was istaken to be. The following forulas are also derived in [G]. Lea 1. Let u be a sooth function on a closed pseudoheritian anifold of diension + 1. We have the following integral equalities 1 u β u β,0 u β u β,0 = u α,β uα,β R ασ u σ u α, 1 1 u β u β,0 u β u β,0 = bu 4 uα,α 1 A αβ u α u β A αβ u α u β, 1 u β u β,0 u β u β,0 = 4 u α,β b u + 1 A ασ u α u σ A ασ u σ u α. We can now state the ain estiate on λ 1. For copletion and future application in the next section, we also provide the detail of the proof here. Theore 7. Let be a closed pseudoheritian anifold of diension Suppose for any X T 1,0 3.1 Ric X, X + 1 T or X, X κ X, where κ is a positive constant. Then the first eigenvalue of b satisfies λ κ.

6 6 SONG-YING LI AND XIAODONG WANG Reark 4. In ters of our local unitary frae, the assuption 3.1 eans that for any X = f α T α R ασ f σ f α [Aασ f α f σ A ασ f σ f α ] κ f α. α Proof. Suppose b u = λ 1 u. Applying the Bochner forula, we have 0 = u α,β + u α,β λ1 b u +R ασ u σ u α + 1 [Aασ u α u σ A ασ u σ u α ] + 1 u β u β,0 u β u β,0. We write the last ter as c ties the first identity plus 1 c ties the second identity of Lea 1, 0 = u α,β + u α,β λ1 b u = +R ασ u σ u α + 1 [Aασ u α u σ A ασ u σ u α ] + c uα,β u α,β R ασ u σ u α 1 c + λ 1u 4 1 c uα,α 1 c 1 A αβ u α u β A αβ u α u β 1 c u α,β c uα,β 1 c λ 1 b u + 1 c 1 R ασ u σ u α c [Aασ u α u σ A ασ u σ u α ] 4 1 c uα,α. Since α,β=1 u α,β uα,α /, we have 0 1 c u α,β 1 c λ 1 b u + 1 c 1 R ασ u σ u α c [Aασ u α u σ A ασ u σ u α ] + [ c ] c uα,α. We choose c such that 1 + c c = 0, i.e. c = 3/ 4 +. Then u α,β λ 1 b u { 1 + R ασ u σ u α } 1 [Aασ u α u σ A ασ u σ u α ]. + 1

7 AN OBATA-TYPE THEORE IN CR GEOETRY 7 Therefore 1 κ λ 1 b u + u α,β 0. It follows that λ 1 κ/ + 1 when. 4. Equality case We now discuss the equality case. By scaling, we can assue κ = + 1 / and thus λ = /. Proposition 4. If equality holds in Theore 7, we ust have u α,β = 0, u α,β = u + u 0 δ αβ, 1 u 0,α = A ασ u σ + u α, u 0,0 = 1 4 u + 4 IA ασ,αu σ. oreover, at any point where b u Aαβ = Q b u 4 u αu β, where Q = 1A ασ u α u σ. Proof. If equality holds, we ust have u α,β = 0 and 4.6 u α,β = fδ αβ, where f is a coplex-valued function. Taking conjugate of 4.6 yields fδ αβ = u α,β = u β,α 1δ αβ u 0 = f 1u 0 δαβ. Hence If = 1 u 0. We also have u = bu Thus Ref = 1 4u. Therefore = u α,α u α,α = f + f. 4.7 f = 1 4 u + 1 u 0, This proves 4.. Differentiating 4.1, we have Therefore 0 = u α,βγ u α,γβ = 1 A αγ u β A αβ u γ. A αγ u β A αβ u γ = 0.

8 8 SONG-YING LI AND XIAODONG WANG Fro this we easily obtain 4.5. Differentiating 4.6, we have f γ δ αβ = u α,βγ = u α,γβ + 1 h αβ A σ γ h αγ A σ u β σ = f β δ αγ + 1 δ αβ A σ γ δ αγ A σ u β σ, i.e. f γ 1Aγ σu σ δαβ = f β 1A σu β σ δ αγ. It follows f γ 1A σ γ u σ = 0. Using 4.7, this yields 4.8 1Aασ u σ = 1 u α + 1u α,0. This then iplies 4.3 by using the first identity of Proposition 3. To prove the last identity, taking trace of 4. we obtain u + u 0 = u α,α. Differentiating and using Proposition 3 yields u 0 + u 0,0 = u α,α0 = u α,0α A ασ,α u σ 1 = A ασ u σ + u α A ασ,α u σ,α 1 = A ασ,α u σ + u α,α A ασ,α u σ = 1 1IA ασ,α u σ u + u 0 = 4 u IA ασ,α u σ 8 u. Taking the iaginary part yields 4.4. Lea. We also have 4.9 R ασ u σ A ασ u σ = + 1 u α. Proof. This follows easily fro the fact that equality is achieved by X = u σ T σ in 3.1. We can also derive it in the following way. Differentiating 4.1 and using 4.6 yields 0 = u α,βγ = u α,γβ + 1δ βγ u α,0 R βγασ u σ = u β + u 0,β δ αγ + 1u α,0 δ βγ R βγασ u σ.

9 AN OBATA-TYPE THEORE IN CR GEOETRY 9 Taking trace over β and γ, we obtain 0 = u α + u 0,α + 1u α,0 + R ασ u σ = u α + A ασu σ uα,0 + R ασ u σ + 1 = u α A ασ u σ + R ασ u σ, where in the last step we have used 4.8 to replace u α,0. Lea 3. Q is real and nonpositive. Reark 5. This lea will not be needed in the proof of the rigidity. However, it yields a quick proof if we assue the following extra condition A αβ,αβ = 0, i.e. the double divergence of the torsion is zero. Indeed, integrating by parts and using 4.1 we obtain Q = 1 A αβ,α u β u 1 = A αβ,αβ u = 0. As Q is nonpositive, this iplies that Q = 0. Therefore A = 0. See the discussion on the torsion-free case below. Proof. Fro 4.9 we have R ασ u α u σ Q = + 1 b u. This shows that Q is real. Taking conjugate, we also have Q = 1A ασ u α u σ. In the inequality 3.1, taking X = e it u α T α yields R ασ u α u σ Q cos t + 1 b u. Therefore Q 0. Theore 4 follows fro Lea 4. The torsion A = 0. The proof of this stateent will be presented in the next section. Assuing this lea, Theore 4 then follows fro Chang and Chiu [CC1]. In the following, we present a sipler and ore direct arguent. Since A vanishes, we have u 0,α = u α,0 = u α, u α,β = 0, u 0,0 = 1 4 u u α,β = u + u 0 δ αβ.

10 10 SONG-YING LI AND XIAODONG WANG By Proposition, we obtain Proposition 5. Let D u be the Hessian of u with respect to the Rieannian etric g θ. Then A = D u = 1 4 ug θ. By Obata s theore Theore,, 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 this can be easily proved by the first forula in Reark 1. 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 0 a 1 a 1 0 where a i 0. Therefore T = i a i Since T is of unit length we ust have 0 a+1 a +1 0 y i x i x i y i 4 i a i x i + y i = 1 on S +1. Therefore all the a i s are equal to 1/. It follows that This finishes the proof of Theore 4. Let ψ = A. In local unitary frae θ = g 0 T, = 1 z Proof of Lea 4 ψ = Aαβ. We note that ψ is continuous and ψ is sooth. Let K = { b u = 0}. By 4.5, on \K ψ is sooth and ψ = Q b u, or A αβ = 1ψ u αu β b u.

11 AN OBATA-TYPE THEORE IN CR GEOETRY 11 Lea 5. The copact set K is of Hausdorff diension at ost n n = + 1 = di. ore precisely we have a countable union K = i=1 E i, where each E i has finite n diensional Hausdorff easure: H n E i <. Reark 6. Here the Hausdorff diension is defined using the distance function of the Rieannian etric g θ. Proof. We have K = K 1 K, where K 1 = {p K : u p 0 or u 0 p 0}, K = {p : u p = 0 and du p = 0}. We first prove that K 1 is of Hausdorff diension n. Suppose p K 1. In a local unitary frae {T α } we have by Proposition 4 u α,β = u + u 0 δ αβ. We write T α = X α 1Y α in ters of the real and iaginary parts. Then we have real local vector fields {Z i }, where Z i = X i for i and Z i = Y i for i >. Along K 1 the above equation takes the following for X β X α u + Y β Y α u = 1 4 uδ αβ, Y β X α u X β Y α u = 1 u 0δ αβ. Since either u p or u 0 p 0, fro the above equation it is straightforward to check that there exists i < j such that the local ap F : q Z i u q, Z j u q fro to R is of rank at p. By the iplicit function theore, F 1 0 is a codiension subanifold at p. As K 1 F 1 0, we conclude that K 1 is of Hausdorff diension at ost n. To handle K, we note that u satisfies the following nd order elliptic equation by Proposition 4 u = + 1 u IA ασ,αu σ. As K is the singular nodal set of u, we have see, e.g. [HHL] Lea 6. We have on \K H n K <. Re ψ α u α = 0. Proof. Let v = T u = u 0. By the second forula of Lea v β v β,0 v β v β,0 = bv 4 v α,α A αβ v α v β A αβ v α v β. We will use Proposition 4 to siplify both sides. On \K v α = u 0,α = 1 ψ + 1 u α, v α,β = 1ψ β u α + ψ u 0 4 u δ αβ.

12 1 SONG-YING LI AND XIAODONG WANG Differentiating the first equation yields v α,β = 1ψ β u α. As v α,β = v β,α, we have ψ β u α = ψ α u β. As a result, on \K there are sooth real functions a, b such that A siple calculation shows 1 v β v β,0 v β v β,0 ψ α = a + ib u α. = 1 ψ + 1 u β u β,0 u β u β,0 = ψ + 1 ψ + 1 b u. The integrand of the right hand side can be siplified as following 1 bv 4 vα,α 1 A αβ v α v β A αβ v α v β = 4 I v α,α 1 A αβ v α v β A αβ v α v β = 4 a b u ψ + 1 u 4 = 4 ψ + 1 ψ b u ψ + 1 u + 4a b u ψ + 1 u 16 a b u 4 ψ + 1 ψ b u. Integrating by parts see the reark below yields ψ + 1 u 4 = Re ψ + 1 u α,α u = ψ + 1 b u 4Re ψ + 1 uψ α u α = ψ + 1 b u 4 ψ + 1 au b u. Plugging these calculations in 5.1, we obtain 16 a b u 4 = 0. Therefore Re ψ α u α = a b u = 0. Reark 7. We can justify the integration by parts in the following way. We note that the copact set K has zero -capacity by Lea 5 cf. [EG, HK]. Therefore there exists a sequence χ k Cc \K s.t. χ k 1 in W 1,. Then φ + 1 u α,α u χ k = φ + 1 b u χ k + φ + 1 φ α u α u χ k + E k, where E k = φ + 1 u α uχ k χ k α.

13 AN OBATA-TYPE THEORE IN CR GEOETRY 13 It is easy to see that li k E k = 0. Therefore, letting k yields φ + 1 u α,α u = φ + 1 b u + 4 φ + 1 φ α u α u. We now prove Lea 4. Suppose ψ is not identically zero. Let ε be a regular value of ψ such that {ψ ε} is a nonepty doain with sooth boundary. Define { ψ ψ ε, if ψ ε; F = 0 if ψ < ε. Then F W 1,. Integrating by parts, we obtain F u k+1 = 4 Re F u k+1 u α,α 4 k + 1 = Re F u k b u ψ 4εψ + ε u k+1 Reu α ψ α {ψ ε} 4 k + 1 = F u k b u, by Lea 6. Integrating by parts again, we have F u k b u = u k ψ ε + ψ bu = Re 1 u k ψ ε + A αβu α u β 1 = Re ψ ε + k + 1 A αβ u k+1 u α β 1 = Re ψ ε + k + 1 A αβ,αu k+1 u β + 1 = Re ψ ε + k + 1 A αβ,αu k+1 u β k = Re ψ ε + k + 1 A αβ,αu k+1 u β, {ψ ε} ψ ε u k+1 ψ α A αβ u β {ψ ε} ψ ε ψu k+1 Reψ α u α by Lea 6 again. Let C be super nor of diva = A αβ,α. Then by the Hölder inequality F u k b u C F u k+1 b u ε k + 1 1/ 1/ C F u k+1 F u k b u. ε k + 1

14 14 SONG-YING LI AND XIAODONG WANG Hence F u k b u [ C ε k + 1 4C ] ε k + 1 F u k+1 F u k b u, 4C where in the last step we used 5.. Choosing k such that ε k+1 1 yields F u k b u = 0. This is a contradiction. Therefore Lea 4 is proved. Inspecting the proof of the rigidity, it is clear that we only need to have a nonconstant function u satisfying 4.1 and 4. as all the other identities used in the proof are derived fro these two. In suary, we have proved the following theore. Theore 8. Let be a closed pseudoheritian anifold of diension Suppose there exists a non-constant function u C satisfying u α,β = 0, u α,β = u + u 0 δ αβ. Then is equivalent to the sphere S +1, 1 z 1. This is equivalent to Theore 5 by scaling. 6. Rearks for the case + 1 = 3 Generally speaking, 3-diensional CR anifolds are ore subtle to understand than higher diensional ones. A faous exaple is the CR ebedding proble. In our situation, it is not clear if Theore 8 is true in 3 diensions. The reason is that 4.3 does not follow fro 4.1 and 4. in 3 diensions In deriving 4.8 we need at least indices. The arguents in Section 5 do yield the following weaker rigidity theore in diension 3 with 4.3 as an extra condition. Theore 9. Let be a 3-diensional closed pseudoheritian anifold. Suppose there exists a non-constant function u C satisfying u 1,1 = 0, u 1,1 = u + u 0, 1 u 0,1 = A 11 u 1 + u 1. Then is equivalent to the sphere S 3, 1 z 1. In fact, the eigenvalue estiate Theores 3 is not known in the 3-diensional case without any extra condition. Chang and Chiu in [CC] proved the estiate under the extra condition that the Panietz operator is nonnegative. They also proved that is CR equivalent to the sphere if equality holds and the torsion is zero.

15 AN OBATA-TYPE THEORE IN CR GEOETRY 15 Recall that the Panietz operator P 0 acting on functions on a pseudoheritian anifold of diension + 1 is defined by P 0 u = 4 Re u β,βα + 1A αβ u β.,α It is proved by Graha and Lee [GL] that P 0 is always nonnegative if is closed and of diension 5 in the sense up 0 u 0 for any sooth function u. In 3 diensions, P 0 is known to be nonnegative if the torsion is zero. With our ethod, we can reove the torsion-free condition in the characterization of the equality case in Chang and Chiu s work. Theore 10. Let 3, θ be a closed pseudoheritian anifold such that for any X = ct R 11 c 1A 11 c A 11 c c. If P 0 0, then λ 1 1 and the equality holds if and only if 3, θ is equivalent to S 3, 1 z 1. Reark 8. The first part of the theore was proved by Chiu [Ch] and the second part of the theore was proved by Chang and Chiu [CC] under the extra condition that is torsion free. We sketch the proof here. By Lea. in [CC1], one has u 0 = b u 1A 11 u 1 A 11 u 1] 1 Using Proposition 3 and integrating by parts yields up 0 u. [ 1 ] Re u 1 u 1,0 u 1 u 1,0 = u 0 1A11 u 1 A 11 u 1. Let u be a non-zero first eigenfunction, b u = λ 1 u. Then b u = λ 1 b u.

16 16 SONG-YING LI AND XIAODONG WANG By the Bochner forula Theore 6, Lea 1 and the above two identities, we have 0 = u 1,1 + u 1,1 λ 1 u + R 11 u 1 u [A11 u 1 A 11 u 1] + Re 1 u 1 u 1,0 u 1 u 1,0 = u 1,1 + λ 1u + u λ 1 u + R 11 u [A11 u 1 A 11 u 1] u 0 1A11 u 1 A 11 u 1 = u 1,1 λ 1 u + R 11 u 1 1 1[A11 u 1 A 11 u 1] 3 u 0 4 = u 1,1 λ 1 u + R 11 u 1 1 1[A11 u 1 A 11 u 1] 3 [λ 1 u 1[A 11 u 4 1 A 11 u 1] 1 up 0u] = u 1,1 + 3 up 0u + [ λ 1 u + u ] [ λ 1 u + R 11 u 1 + 1[A 11 u 1 A 11 u 1] This iplies that λ 1 1. If equality holds, we ust have 6. u 1,1 = 0, u 1,1 = u u R 11 u 1 + 1A 11 u 1 A 11 u 1 = u 1. Writing A 11 = e 1θ 1 A 11, u 1 = e 1θ u 1 and X = e 1θ X, by 6.1, we have 6.4 R 11 X + 1 A 11 X e 1θ 1+θ +θ e 1θ 1+θ +θ X, for all θ [0, π. Choosing θ = θ 1 + π, one has 6.5 R 11 X A 11 X X. Therefore, by coparing 6.3 and 6.5 with X = u 1 T 1, Notice that 1A11 u 1 A 11 u 1 = A 11 u 1, A 11 = u 1 1 A 11 u 1. R 11 u 1 + tx A 11 u 1 + tx u 1 + tx 0, on, and equality holds at t = 0. Therefore we obtain by differentiating at t = 0 R 11 u 1 + 1A 11 u 1 = u 1.

17 AN OBATA-TYPE THEORE IN CR GEOETRY 17 Using the 7th forula of Proposition 3, we have u u 0,1 = u 1,11 u 1,11 = 1u 1,0 R 11 u 1 = 1u 0,1 + 1A 11 u 1 R 11 u 1 = 1u 0, A 11 u 1 u 1 Therefore, 1 u 1 + 1u 0,1 = 1 1A 11 u 1, u 0,1 = u 1 + A 11 u 1. Applying Theore 9, the proof of Theore 10 is coplete. References [CC1] S. Chang and H. Chiu, Nonnegativity of CR Paneitz operator and its application to the CR Obata s theore. J. Geo. Anal , no., R [CC] S. Chang and H. Chiu, On the CR analogue of Obata s theore in a pseudoheritian 3-anifold. ath. Ann , no. 1, R [CW] S. Chang and C. Wu, The diaeter estiate and its application to CR Obata s theore on closed pseudoheritian n+1-anifolds. Trans. Aer. ath. Soc , no. 7, R [C] I. Chavel, Eigenvalues in Rieannian Geoetry, Acadeic Press, Inc., New York, London, Toronto, Tokyo, R [Ch] H. Chiu, The sharp lower bound for the first positive eigenvalue of the sublaplacian on a pseudoheritian 3-anifold. Ann. Global Anal. Geo , no. 1, R [DT] S. Dragoir and G. Toassini, Differential geoetry and analysis on CR anifolds. Birkhauser, 006. R [EG] L. C. Evans and R. F. Gariepy, easure theory and fine properties of functions. CRC Press, 199. R [G] A. Greenleaf, The first eigenvalue of a sub-laplacian on a pseudoheritian anifold. Co. Partial Differential Equations , no., R [GL] C. R. Graha and J.. Lee, Sooth solution of degenerate Laplacians on strictly pseudoconvex doains. Duke ath. Jour , no. 3, R [H] Xiaojun Huang, On an n-anifold in the coplex n-space near an elliptic coplex tangent, Jour. of Aer. ath. Soc. Vol 11, 1998, R [HHL] Q. Han, R. Hardt and F.H. Lin, Geoetric easure of singular sets of elliptic equations. Co. Pure Appl. ath , no. 11-1, R [HK] J. Heinonen, T. Kilpeläinen and O. artio, Nonlinear potential theory of degenerate elliptic equations. Dover Publications, 006. [Lee] John. Lee, Pseudo-Einstein structures on CR anifolds, Aer. J. ath , R [L] A. Lichnerowicz, Géoétrie des groupes de transforations, Dunod, Paris R [LL] S.-Y. Li and H.-S. Luk, The sharp lower bound for the first positive eigenvalue of a sub- Laplacian on a pseudoheritian anifold. Proc. Aer. ath. Soc , no.3, R [LT] S.-Y. Li and y-an Tran, On the CR-Obata theore and soe extreal probles associated to pseudoscalar curvature on the real ellipsoids in C n+1. Trans. Aer. ath. Soc., , no. 8, R [O]. Obata, Certain conditions for a Rieannian anifold to be isoetric with a sphere. J. ath. Soc. Japan R [IV] S. Ivanov and D. Vassilev, An Obata type result for the first eigenvalue of the sub-laplacian on a CR anifold with a divergence free torsion, arxiv:

18 18 SONG-YING LI AND XIAODONG WANG [SY] R. Schoen and S.-T. Yau, Lectures on differential geoetry. International Press, Cabridge, A, R [T] N. Tanaka, A differential geoetric study on strongly pseudo-convex anifolds. Lectures in atheatics, Departent of atheatics, Kyoto University, No. 9. Kinokuniya Book-Store Co., Ltd., Tokyo, [W] S.. Webster, Pseudoheritian geoetry on a real hypersurface, J. Diff. Geo., , R Departent of atheatics, University of California, Irvine, CA 9697; School of atheatics and Coputer Science, Fujian Noral University, Fuzhou, China E-ail address: sli@ath.uci.edu Departent of atheatics, ichigan State University, East Lansing, I 4884 E-ail address: xwang@ath.su.edu