PSEUDOHERMITIAN 3-MANIFOLD
SHU-CHENG CHANG1 AND HUNG-LIN CHIU2
Abstract. In this paper, we …rst prove the CR analogue of M. Obata’s theorem on a closed pseudohermitian 3-manifold with free pseudohermitian torsion. Secondly, instead of free torsion case, we have the CR analogue of Li-Yau’s eigenvalue esti-mate on the lower bound estiesti-mate of psitive …rst eigenvalue of the sub-Laplacian in a closed pseudohermitian 3-manifold with nonnegative CR Paneitz operator P0.
Fi-nally, we have a criterion for the positivity of …rst eigenvalue of the sub-Laplacian on a complete noncompact pseudohermitian 3-manifold with nonnegative CR Paneitz operator. The key step is a discovery of integral CR analogue of Bochner formula which involving the CR Paneitz operator.
1. Introduction
A. Greenleaf ([Gr]) proved the pseudohermitian analogue of Lichnerowicz’s Theo-rem ([L]) for the lower bound (n=n + 1)k0 of the …rst positive eigenvalue 1 of the
sublaplacian for a pseudohermitian manifold M2n+1 with n 3and under a condition
on the Webster curvature and the torsion as follows (see the de…nitions and notations in next section):
(1.1) Ricm(Z; Z) + (n=2)T orm(Z; Z) k0hZ; ZiL
for all m 2 M, Z 2 T1;0, and for some positive constant k0.
In [LL], S.-Y. Li and H.-S. Luk proved the same result for the cases n = 1; n = 2. However, in the case n = 1, they needed a condition depending not only on the Webster curvature and the pseudohermitian torsion, but also on a covariant derivative
1991 Mathematics Subject Classi…cation. Primary 32V05, 32V20; Secondary 53C56.
Key words and phrases. Lichnerowicz-Obata Theorem, Pseudohermitian manifold, Webster metric, Tanaka-Webster curvature, Pseudohermitian torsion, CR Paneitz operator, Sub-Laplacian, Carnot-Carathéodory distance, Diameter.
Research supported in part by the NSC of Taiwan.
of the torsion : (1.2) Ricm(Z; Z) + 1 2T orm(Z; Z) 3 k0 Bm2(Z; Z) k0hZ; ZiL
for all m 2 M, Z 2 T1;0 and for Z = x1Z1
(1.3) Bm2(x1Z1; x1Z1) = 2jA11j2 x1 2 Re A11;0x1x1 :
Recently, it was proved by the second author that the same result ([Ch]) holds on a pseudohermitian 3-manifold (M3; J; ) under more geometric condition which
involving the positivity of the CR Paneitz operator P0 (see de…nition below) with
respect to (J; ):
We …rst recall the de…nition of CR Paneitz operator P0:We de…ne
P ' = ('111 + iA11'1) 1 = P ' = (P1') 1;
which is an operator that characterizes CR-pluriharmonic functions. Here P1' =
'111+ iA11'1 and P ' = (P1) 1, the conjugate of P . The CR Paneitz operator P0 is
de…ned by
(1.4) P0' = 4 b(P ') + b(P ') ;
where b is the divergence operator that takes (1; 0)-forms to functions by b( 1 1) = 1;1, and similarly, b( 1 1) = 1;1. We observe that
(1.5) Z hP ' + P '; db'iL d = 1 4 Z P0' ' d
with d = ^ d : One can check that P0 is self-adjoint, that is, hP0'; i = h'; P0 i
for all smooth functions ' and . For the details about these operators, the reader can make reference to [GL], [H], [L1], [GG] and [FH].
De…nition 1.1. 1. On a complete pseudohermitian 3-manifold (M; J; ); we call the Paneitz operator P0 with respect to (J; ) essentially positive if there exists a constant
> 0 such that (1.6) Z M P ' 'd Z M '2d :
for all real C1 smooth functions ' 2 (ker P
0)? (i.e. perpendicular to the kernel of
P0 in the L2 norm with respect to the volume form d = ^ d ): We say that P0 is
nonnegative if Z
M
P ' 'd 0 for all real C1 smooth functions.
2. We say that (M; J ) has a transversal symmetry if M admits a one-parameter group of CR automorphisms transverse to the holomorphic tangent bundle.
Remark 1.1. 1. The essentially positivity of P0 is a CR invariant in the sense that
it is independent of the choice of the contact form .
2. Let (M; J; ) be a closed pseudohermitian 3-manifold with free torsion. Then the corresponding CR Paneitz operator is essentially positive ( [CCC], [CaC]).
3. If (M; J ) has a transversal symmetry, then there is a contact form such that the corresponding pseudohermitian torsion is zero ([GL]).
Proposition 1.1. ([Ch]) Let (M; J; ) be a closed three-dimensional pseudohermitian manifold with nonnegative Paneitz operator P0: Suppose that
(1.7) Ricm(Z; Z) T orm(Z; Z) k0hZ; ZiL ;
for all m 2 M, Z 2 T1;0, and for some positive constant k0. Let 1 be the …rst positive
eigenvalue with respect to b. Then
1
k0
2 > 0:
As a consequence, the same result holds for a closed three-dimensional pseudo-hermitian manifold with
A11 = 0 and W k0:
Let (S3; bJ ; b) be a sphere S3 with the induced natural CR structure from C2 and the standard contact form b, one can show that ([Ch], [CCC])
1 =
k0
2:
Here k0 is the positive constant Webster curvature of S3: Thus we get the sharp
Then it is natural to conjecture the CR analogue of M. Obata’s Theorem ( [O]) on a closed three-dimensional pseudohermitian manifold (M; J; ):
Conjecture 1.2. Let (M; J; ) be a closed three-dimensional pseudohermitian man-ifold with
A11= 0 and W k0
for a positive constant k0. Suppose that 1 =
k0
2 :
Then (M; J; ) is the standard pseudohermitian 3-sphere (S3; bJ ; b) with constant
Web-ster curvature k0 .
In this paper, we will con…rm the Conjecture 1.2 (Corollary 1.4). In fact, we will have a generalization of Lichnerowicz and Obata Theorem (Theorem 1.3).
We …rst de…ne the Levi metric h on ker by h(X; Y ) = d (X; JY ):
A family of Webster (adapted) metrics h of (M; J; ) are the Riemannian metrics h = h + 2 2; > 0:
Let 1 be the …rst positive eigenvalue of the Laplacian with respect to the metric h . We denote that max jA11j = 0: Here is the main Theorem in the present
paper.
Theorem 1.3. Let (M; J; ) be a closed three-dimensional pseudohermitian manifold with nonnegative Paneitz operator P0: Then
1 (2 + 2
1+ 2 2
0) 1:
As our …rst consequence, one obtains a CR analogue of Obata’s Theorem.
Corollary 1.4. Let (M; J; ) be a closed three-dimensional pseudohermitian manifold with
A11 = 0 and W 2 2:
If 1 = 2
; then (M; J; ) is the standard pseudohermitian 3-sphere (S3; bJ ; b) with constant Webster curvature 2 2.
A piecewise smooth curve : [0; 1] ! M is said to be horizontal if 0(t) 2
whenever 0(t)exists. The length of is then de…ned by l( ) =
Z 1 0
h( 0(t); 0(t))12dt:
The Carnot-Carathéodory distance dc between two points p; q 2 M is de…ned by
dc(p; q) =inf fl( )j 2 Cp;qg ;
where Cp;q is the set of all horizontal curves which join p and q. By Chow connectivity
theorem [Cho], there always exists a horizontal curve joining p and q, so the distance is …nite. We say M is complete if it is complete as a metric space.
Here we recall a result of the authors’previous paper.
Proposition 1.5. ([CC]) Let (M; J; ) be a closed three-dimensional pseudohermitian manifold which has a transversal symmetry. Suppose that
(a)
Ricm(Z; Z) T orm(Z; Z) k0hZ; ZiL ;
for some nonnegative constant k0 and
(b) b(ker P0) ker P0 . Then 1 max 8 < : 1 +p1 + 2(k0+ 0)D2 6D2 e 1+p1+2(k0+ 0)D2 ; k0+ p k2 0 + 6 4 9 = ;: Here 0 = maxjA11j and D is the diameter of (M; J; ). is the constant in (1.6).
Remark 1.2. 1. In Proposition 1.5, if A11 = 0 ; then (b) is satis…es and then we
have the Proposition 1.1 for k0 = 0 ([CC]).
2. For a closed pseudohermitian 3-manifold (M; J; ) with its Webster metric h , if we assume that the Ricci curvature w.r.t.h
Ric 2k
for some positive constant k; then by Li-Yau estimate ([LY])
1
expf [1 + (1 + 2c2D2k)12]
g cD2
Then, for the second consequence of Theorem 1.3, one can have a CR analogue of Li-Yau eigenvalue estimate ([LY]) on a closed pseudohermitian 3-manifold with nonnegative CR Paneitz operator P0which is served as a generalization of Proposition
1.5 :
Corollary 1.6. Let (M; J; ) be a closed pseudohermitian 3-manifold with nonneg-ative CR Paneitz operator P0. Assume that there exist positive constants k1; k2 with
W k1 and k2 = maxfjA11j; jA11;0j 1 2;jA11; 1j 2 3g:
Then for a positive constant < 1 with 2k 1 and k = maxfk1; k2g;either (i) 2 1 or (ii) 1 expf [1 + (1 + 7c2D2 2)12]g 5cD2
where D is the diameter of (M; J; ):
Remark 1.3. In Corollary 1.6, the pseudohermitian torsion is nonzero which is the main di¤erent from our previous works ([Ch], [CC]).
Third, we have a criterion for the positivity of …rst eigenvalue of the sub-Laplacian on a complete noncompact pseudohermitian 3-manifold with nonnegative CR Paneitz operator. In fact from Corollary 2.3, the result of Theorem 1.3 holds also on a com-plete noncompact pseudohermitian 3-manifold with nonnegative CR Paneitz operator P0:
Theorem 1.7. Let (M; J; ) be a complete noncompact pseudohermitian 3-manifold with nonnegative CR Paneitz operator P0: Then
1 (2 + 2
1+ 2 2
Corollary 1.8. Let (M; J; ) be a complete noncompact pseudohermitian 3-manifold with nonnegative CR Paneitz operator P0: Suppose that 1 > 0 on (M; h ) for some
> 0; then
1 > 0
on (M; J; ).
We brie‡y describe the methods used in our proofs. In section 2; we …rst give a brief introduction to pseudohermitian geometry and then derive our crucial integral CR analogue of Bochner formula which involving CR Paneitz operator (Lemma 2.2). In section 3; we obtain the relations between the Riemannian Ricci tensors with respect to the Webster metric and pseudohermitian-Ricci tensors. Finally, by comparing the Laplacian w.r.t. Webster metric and Sub-Laplacian, the proofs of main theorems on CR analogue of Obata’s Theorem (Corollary 1.4) and Li-Yau’s eigenvalue estimate (Corollary 1.6) are complete as in section 4.
Acknowledgments. The …rst author would like to express his thanks to Prof. S.-T. Yau for constant encouragement and Prof. J.-P. Wang for valuable discussions.
2. CR Analogue of Bochner Formula and Paneitz Operator We …rst give a brief introduction to pseudohermitian geometry (see [L1], [L2] for more details). Let M be a closed 3-manifold with an oriented contact structure . There always exists a global contact form , obtained by patching together local ones with a partition of unity. The characteristic vector …eld of is the unique vector …eld T such that (T ) = 1 and LT = 0or d (T; ) = 0. A CR structure compatible
with is a smooth endomorphism J : ! such that J2 = Id. A pseudohermitian
structure compatible with is a CR-structure J compatible with together with a global contact form . The CR structure J can extend to C and decomposes C into the direct sum of T1;0 and T0;1 which are eigenspaces of J with respect to
i and i, respectively.
Let fT; Z1; Z1g be a frame of T M C, where Z1 is any local frame of T1;0; Z1 =
Z1 2 T0;1 and T is the characteristic vector …eld. Then
n
; 1; 1o, the coframe dual to fT; Z1; Z1g, satis…es
for some positive function h11. Actually we can always choose Z1 such that h11= 1;
hence, throughout this paper, we assume h11= 1
The Levi form h ; iL is the Hermitian form on T1;0 de…ned by
hZ; W iL = i d ; Z^ W :
We can extend h ; iL to T0;1 by de…ning Z; W L =hZ; W iL for all Z; W 2 T1;0.
The Levi form induces naturally a Hermitian form on the dual bundle of T1;0, denoted
by h ; iL , and hence on all the induced tensor bundles. Integrating the Hermitian
form (when acting on sections) over M with respect to the volume form d = ^ d , we get an inner product on the space of sections of each tensor bundle. We denote the inner product by the notation h ; i. For example
(2.2) h'; i =
Z
M
' d ; for functions ' and .
The pseudohermitian connection of (J; ) is the connection r on T M C (and extended to tensors) given in terms of a local frame Z1 2 T1;0 by
rZ1 = 11 Z1; rZ1 = 11 Z1; rT = 0;
where 11 is the 1-form uniquely determined by the following equations:
d 1 = 1^ 11+ ^ 1; 1 0 mod 1;
0 = 11+ 11;
(2.3)
where 1 is the pseudohermitian torsion. Put 1 = A1
1 1. The structure equation
for the pseudohermitian connection is
(2.4) d 11 = W 1^ 1+ 2iIm(A11;1 1^ );
where W is the Tanaka-Webster curvature.
We will denote components of covariant derivatives with indices preceded by comma; thus write A1
fT; Z1; Z1g. For derivatives of a scalar function, we will often omit the comma,
for instance, '1 = Z1'; '11= Z1Z1' 11(Z1)Z1'; '0 = T 'for a (smooth) function.
For a real function ', the subgradient rb is de…ned by rb'2 and hZ; rb'iL =
d'(Z) for all vector …elds Z tangent to contact plane. Locally rb' = '1Z1+ '1Z1.
We can use the connection to de…ne the subhessian as the complex linear map
(rH)2' : T1;0 T0;1 ! T1;0 T0;1; by (rH)2'(Z) =rZrb': Also b' = T r (rH)2' = ('11+ '11): For all Z = x1Z 1 2 T1;0, we de…ne Ric(Z; Z) = W x1x1 = WjZj2L ; T or(Z; Z) = 2Re iA11x1x1:
Next we derive the following CR analogue of Bochner formula. Lemma 2.1. For a real function ',
1 2 bjrb'j 2 = j(rH)2' j2+ 3 < rb';rb b' >L +(2Ric 3T or)((rb')C; (rb')C) 4 < P ' + P '; db' >L :
Here (rb')C = '1Z1 is the corresponding complex (1; 0)-vector …eld of rb' and
db' = '1 1+ '
1 1:
Proof. First from [Gr], we have for a real function '
(2.5)
bjrb'j2 = 2j(rH)2'j2+ 2 <rb';rb b' >L
+(4Ric + 2T or)((rb')C; (rb')C)
Then Lemma 2.1 follows from (2.5) and the following (2.6).
(2.6)
< Jrb';rb'0 >L = <rb';rb b' >L
2T or((rb')C; (rb')C)
2 < P ' + P '; db' >L :
For a complete proof of (2.6), we refer to the authors’previous paper [CC].
On the other hand, one can have the integral version of CR Bochner formula. Lemma 2.2. Let (M; J; ) be a closed pseudohermitian 3-manifold. Then
Z M '20 d = Z M ( b') 2 d + 2 Z M T or((rb')C; (rb')C)d 1 2 Z M 'P0' d
for any ' 2 C1(M ): In additional, if the CR Paneitz operator P
0 is nonnegative, then Z M '20 d Z M ( b') 2 d + 2 0 Z Mjr b'j2 d :
Proof. By integrating (2.6) and using (1.5), we have Z '20dV = Z ( b')2dV + 2 Z T or((rb'); (rb'))dV 1 2 Z P0' 'dV: (2.7)
This completes the proof.
Moreover, the results of Lemma 2.2 hold also on a complete pseudohermitian 3-manifold.
Corollary 2.3. Let (M; J; ) be a complete pseudohermitian 3-manifold. Then Z '20 d = Z ( b') 2 d + 2 Z T or (rb')C; (rb')C d 1 2 Z P0' ' d for any ' 2 C1
0 ( ) with M. In additional, if the CR Paneitz operator P0 is
nonnegative, then Z '20 d Z ( b') 2 d + 2 0 Z jrb'j2 d :
3. Curvature Tensors for the Webster Metric
In this section, we derive the relations between the Ricci tensors Rij with respect to the Webster metric h and pseudohermitian-Ricci tensors.
Theorem 3.1. Let Rij denote Ricci curvature tensors with respect to the Webster metric h : (i) If A11= 0; then Rij = 0 B B B @ 2W 2 2 0 0 0 2W 2 2 0 0 0 2 2 1 C C C A:
(ii) If the torsion is nonzero, then
R11= 2W 2 2 2i 2Im A11 11(T ) + 2 Im A11 2T (Re A11) ; R22= 2W 2 2+ 2i 2Im A11 11(T ) 2 Im A11+ 2T (Re A11); R33= 2 2jA11j 2 + 2 2; R12= 2i 2Re A11 11(T ) 2 Re A11 2T (Im A11) ; R13= 2 Re A11;1; R23= 2 Im A11;1:
Proof. Note that we write 11 = i!for some real 1-form ! by (2.3) and Z1 = 12(e1 ie2)
for real vectors e1; e2: It follows e2 = J e1: Let e1 = Re( 1); e2 = Im( 1): Then
fe1; e2; = e3
g is dual to fe1; e2; e3 = Tg:
Now put
!1 = e1; !2 = e2; !3 = 1e3:
Then we have the Riemannian structure equations with respect to h :
d!i = !j ^ !ij; 16 i; j 6 3; 0 = !ji + !ij; d!ji = !ki ^ !jk+1 2Rijkl! k ^ !l; 16 i; j; k; l 6 3: (3.1)
and 1 = !1+ i!2 which satis…es the structure equations as in (2.3) and (2.4).
We …rst discuss the relation of 11; 1 with !ji.
d 1 = d!1+ id!2
= (!1+ i!2)
^ i!2
1+ !3^ (!13+ i!23):
On the other hand,
d 1 = 1^ 11+ ^ 1
= (!1+ i!2)^ 11+ !3^ 1: Hence by Cartan lemma
11 = i!21+ a(! 1+ i!2) + b!3; 1 = !1 3+ i! 2 3+ b(! 1 + i!2) + c!3; (3.2)
for some complex functions a; b; c. Similarly d!3 = !1^ !31+ !2^ !32: But d!3 = 1d = 2 1!1^ !2: Then !31 = A!1+ (B + 1)!2; !32 = (B 1)!1+ D!2; (3.3)
for some real functions A; B; D.
Substitute (3.3) into (3.2), we obtain
1
= [b A i(B 1)]!1+ [ib (B + 1) iD]!2+ c!3 0 mod 1;
and then
On the other hand from
0 = 11+ 11
= (a + a)!1+ (ia ia)!2+ (b + b)!3; we have
(3.5) a = 0 and Re b = 0:
From (3.4) together with (3.5), we get
A + D = 0; a = c = 0 and b = i 1:
These together with (3.2) show that
11 = i(!21 2 ); 1 = 1( A iB) 1: (3.6) However, 1 =A1 1 1 =A
11 1.T husA=- Re A11 and B = Im A11. Substituting
into (3.3) we get !31 = ( Re A11) !1+ Im A11+ 1 !2; !32 = Im A11 1 !1+ ( Re A11) !2: (3.7) Note that (3.8) 2 1 = id 11 = i[W 1^ 1+ 2i Im(A11;1 1^ )] = 2W e1^ e2 + 2 Im(A11;1 1^ ) = 2W e1 ^ e2 + 2 Im A 11;1e1^ e3+ 2 Re A11;1e2^ e3:
Next we compute the relations between the Webster curvature W , the pseudoher-mitian torsion 1 and the curvatures with respect to h :
1 2 = d! 1 2+ 2 d = !32^ !13+ R2112!1 ^ !2+ R2113!1^ !3 + R2123!2^ !3+ 2 2!1^ !2 = 2jA11j 2 + 3 2+ R2112 !1^ !2+ R2113!1^ !3+ R2123!2^ !3: (3.9)
Using (3.7), (3.9) and comparing this with (3.8), we obtain R1212 2jA11j 2 + 3 2 = 2W; R1213 = 2 Im A11;1 ; R1223 = 2 Re A11;1 : (3.10)
Next from (3.1) and (3.7), we have
d!3 1 = d ( Re A11)!1+ ( Im A11+ 1)!2 = ( Re A11) d!1+ ( Im A11+ 1)d!2 +d ( Re A11)^ !1+ d ( Im A 11)^ !2: But d!1 = !2 ^ !1 2+ !3^ !13 = !2 ^ !12+ ( Re A11) !1^ !3 +( Im A11+ 1)!2 ^ !3 and d!2 = !1 ^ !2 1+ !3^ !23 = !1 ^ !21+ ( Re A11) !2^ !3 +( Im A11 1)!1^ !3: Then d!3 1 = ( Re A11) !2 ^ !12+ ( Im A11+ 1)!1^ !21 + 2jA11j2 2 !1 ^ !3+ d ( Re A 11)^ !1+ d ( Im A11)^ !2:
On the other hand,
d!31 = !21^ !32+ X 16i<j63 R13ij!i^ !j = ( Im A11+ 1)!1^ !21 ( Re A11) !2^ !21+X i<j R13ij!i^ !j:
Hence P i<jR13ij!i^ !j = 2 Re A11!2 ^ !1 2 2 Im A11!1^ !21+ 2 jA11j 2 2 !1 ^ !3 +d ( Re A11)^ !1+ d ( Im A11)^ !2: Therefore we get R1313 = 2i 2Im A11 11(T ) 2ImA11+ 2T (Re A11) + 2jA11j 2 2 ; R1323 = 2i 2Re A11 11(T ) + 2 Re A11+ 2T Im A11: (3.11) Similarly we have d!3 2 = ( Re A11) d!2+ ( Im A11 1)d!1+ d ( Re A11)^ !2+ d ( Im A11)^ !1 = ( Re A11) !1^ !21 + ( Im A11 1)!2^ !12+ 2 jA11j 2 2 !2 ^ !3 +d ( Re A11)^ !2 + d ( Im A 11)^ !1:
On the other hand,
d!32 = !12^ !31+P16i<j63R23ij!i^ !j = ( Im A11 1)!2 ^ !1 2+ ( Re A11) !1 ^ !12+ P i<jR23ij!i^ !j: Therefore P i<jR23ij!i^ !j = 2 Re A11!1 ^ !2 1 2 Im A11!2^ !12+ 2 jA11j 2 2 !2 ^ !3 +d ( Im A11)^ !1+ d ( Re A11)^ !2: Thus (3.12) R2323 = 2i 2Im A11 11(T ) + 2 Im A11 2T (Re A11) + 2jA11j2 2 and R1313+ R2323 = 2 2jA11j2 2 2:
All together imply
4W = 2R1212 R1313 R2323+ 4 2:
Together with (3.10), (3.11) and (3.12), we get that the Ricci curvature Rij of M with respect to the Webster metric h is
R11 = 2W 2 2 2i 2Im A11 11(T ) + 2 Im A11 2T (Re A11) ; R22 = 2W 2 2+ 2i 2Im A11 11(T ) 2 Im A11+ 2T (Re A11); R33 = 2 2jA11j 2 + 2 2; R12 = 2i 2Re A11 11(T ) 2 Re A11 2T (Im A11) ; R13 = 2 Re A11;1; R23 = 2 Im A11;1: (3.13) In particular, for A11 = 0 (3.14) Rij = 0 B B B @ 2W 2 2 0 0 0 2W 2 2 0 0 0 2 2 1 C C C A:
This completes the proof.
Remark 3.1. If the torsion is nonzero, one can choose a suitable coordinate with
11(T ) = 0 at a point. It follows that
R11 = 2W 2 2+ 2 Im A11 2T (Re A11) ; R22 = 2W 2 2 2 Im A11+ 2T (Re A11); R33 = 2 2jA11j2+ 2 2; R12 = 2 Re A11 2T (Im A11) ; R13 = 2 Re A11;1; R23 = 2 Im A11;1: (3.15)
4. The Proofs
Let d the distance of h . K. Fukaya ([Fu]) observed that the metric space (M; d ) converges to a Carnot-Carathéodory metric space (M; dc) and converges to a
sub-Laplacian b as ! 0:
For a real function ' and e1 = e1; e2 = e2; e3 = T;we have
b =
1
2('e1e1 + 'e2e2); !3(T ) = 0:
Lemma 4.1. For a real function ', we have
' = 2 b' + 2T2':
Proof. We compute that
' = 3 X j=1 ej(ej') !k j(ej)ek' = 2 b' + e3(e3') !3(e3)e ' !3(e )e3' = 2 b' + 2T2' !13(e1)T ' !32(e2)T ' = 2 b' + 2T2' AT ' + AT ' = 2 b' + 2T2'; A = Re A11:
Proof of Theorem 1.3
:Proof. First note that !1 = e1; !2 = e2; !3 = 1e3: Then
d = !1^ !2^ !3 = 1e1^ e2^ e3 and
d = 1 2
1d :
Therefore from Lemma 4.1 2 Z ' 'd = 2 Z ' b' d + 2 Z 'T2' d and then (4.1) 2 Z jr 'j2d = 2 Z jrb'j2 d + 2 Z '20 d :
Now from (4.1) and Lemma 2.2, it follows that 2 Z jr 'j2d 2 Z jrb'j2 d + 2 Z ( b')2 d + 2 0 Z jrb'j2 d : Suppose that b' = 1': Then 1 = R jrb'j2 d R '2 d and R jr 'j2d R '2d (2 + 2 1 + 2 2 0) 1:
Now by de…nition for the eigenvalue, we have
1 R jr 'j2d R '2d (2 + 2 1+ 2 2 0) 1:
This completes the proof.
De…nition 4.1. We call a CR structure J spherical if Cartan curvature tensor Q11
vanishes identically. Here Q11= 1 6W11+ i 2W A11 A11;0 2i 3A11; _ 11:
Note that (M; J; ) is called a spherical pseudohermitian 3-manifold if J is a spher-ical structure. We observe that the spherspher-ical structure is CR invariant and a closed spherical pseudohermitian 3-manifold (M; J; ) is locally CR equivalent to the stan-dard pseudohermitian 3-sphere (S3; bJ ; b): In additional, if M is simply connected,
then (M; J; ) is the standard pseudohermitian 3-sphere.
Proof of Corollary 1.4
: Proof. It follows from (3.14) thatRij = 0 B B B @ 2W 2 2 0 0 0 2W 2 2 0 0 0 2 2 1 C C C A and then Ric(h ) (3 1) 2 = 2 2:
On the other hand, Lichnerowicz Theorem implies
1 3 2
: It follows from Theorem 1.3 that
3 2 1 (2 + 2 1) 1: Again from Proposition 1.1
1
2:
Hence if 1 = 2; it follows that
1 = 3 2
:
Thus, due to Obata Theorem, M is simply connected and W = 2 2: On the other hand, A11= 0. Then M is spherical as well. All these imply (M; J; ) is the standard
pseudohermitian 3-sphere (S3; bJ ; b).
Proof of Corollary 1.6
:Proof. By our assumptions and (3.15), we have
Ric 7 2 = 2 7
2
2
: Therefore by Li-Yau eigenvalue estimate ([LY]), we have
1
expf [1 + (1 + 7c2D2 2)1 2]g
cD2
for some c depending on the dimension of M and D2=diam(M; h ): But we …rst note
that d (x; x0) is an increasing function of 1 and then
d (x; x0) dc(x; x0)
for the distance d (x; x0) with respect to the metric h : Hence
D D:
It follows that
(2 + 2 1+ 2 2 0) 1
expf [1 + (1 + 7c2D2 2)12]g
Now if 1 < 2 ; then 5 1 (2 + 2 1+ 2 2 0) 1 expf [1 + (1 + 7c2D2 2)1 2]g cD2 : That is 1 expf [1 + (1 + 7c2D2 2)12]g 5cD2 : References
[CCC] S.-C., Chang, J.-H. Cheng and H.-L. Chiu : The Fourth-order Q-curvature ‡ow on a CR 3-manifold, to appear in IUMJ.
[CaC] J. Cao and S.-C., Chang, Pseudo-Einstein and Q-Flat Metrics with Eigenvalue Estimates on CR-Hypersurfaces, Indiana Univ. Math. J. (to appear).
[CC] S.-C. Chang and H.-L. Chiu, On the estimate of …rst eigenvalue of a sublaplacian on a pseudo-hermitian 3-manifold, to appear in Paci…c J. of Math.
[Ch] H.-L. Chiu: The sharp lower bound for the …rst positive eigenvalue of the sublaplacian on a pseudohermitian 3-manifold, Annals of Global Analysis and Geometry (2007), to appear. [Cho] W.-L. Chow : Uber System Von Lineaaren Partiellen Di¤erentialgleichungen erster
Ordu-ung,. Math. Ann. 117 (1939), 98-105.
[Fu] K. Fukaya, Collapsing of Riemannian Manifolds and Eigenvalues of Laplacian Operators, Invent. Math. 87 (1987), 517-547.
[FH] C. Fe¤erman and K. Hirachi, Ambient Metric Construction of Q-Curvature in Conformal and CR Geometries, to appear in Math. Res. Lett.
[GG] A. R. Gover and C. R. Graham, CR Invariant Powers of the Sub-Laplacian, preprint. [GL] C. R. Graham and J. M. Lee, Smooth Solutions of Degenerate Laplacians on Strictly
Pseudo-convex Domains, Duke Math. J., 57 (1988), 697-720.
[Gr] A. Greenleaf: The …rst eigenvalue of a Sublaplacian on a Pseudohermitian manifold. Comm. Part. Di¤ . Equ. 10(2) (1985), no.3 191–217.
[H] K. Hirachi, Scalar Pseudo-hermitian Invariants and the Szegö Kernel on 3-dimensional CR Manifolds, Lecture Notes in Pure and Appl. Math. 143, pp. 67-76, Dekker, 1992.
[L] A. Lichnerowicz, Géometrie des groupes de transformations, Dunod, Pairs, 1958.
[LY] P. Li and S.-T. Yau, Estimates of Eigenvalues of a Compact Riemannian Manifold, AMS Proc. Symp. in Pure Math. 36 (1980), 205-239.
[L1] J. M. Lee, Pseudo-Einstein Structure on CR Manifolds, Amer. J. Math. 110 (1988), 157-178. [L2] J. M. Lee, The Fe¤erman Metric and Pseudohermitian Invariants, Trans. A.M.S. 296 (1986),
[LL] S.-Y. Li and H.-S. Luk : The sharp lower bound for the …rst positive eigenvalue of a sub-Laplacian on a Pseudo-Hermitian manifold. proc. Amer. Math. Soc. 132 (2004), no.3 789– 798.
[O] M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333-340.
[P] S. Paneitz, A Quartic Conformally Covariant Di¤erential Operator for Arbitrary Pseudo-Riemannian Manifolds, preprint, 1983.
[SY] R. Schoen & S.-T. Yau, Lectures on Di¤erential Geometry, International Press, 1994. [W] J.-P. Wang, Private communication, 2005.
1Department of Mathematics, National Tsing Hua University, Hsinchu 30013,
Tai-wan, R.O.C.
E-mail address: [email protected]
2Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li
32023, Taiwan, R.O.C.
