CARTAN FLOW ON A CR 3-MANIFOLD
CHIN-TUNG WU
Abstract. In this note we derive a formula for the derivative of the CR Yamabe constant Y(J(t)); where J(t) is a solution of the Cartan ‡ow on a closed CR
3-manifold. We also give a simple application.
In this paper, following the way of Chang and Lu [CLu], we derive a formula for the derivative of the CR Yamabe constant Y(J(t)); where J(t) is a solution of the
Cartan ‡ow on a closed CR 3-manifold. As an application, we show that if (J(0); )
is of nonnegative constant Tanaka-Webster curvature and the real part (or imagine part) of torsion along T -direction derivative vanishes for the initial data, then the Cartan ‡ow will increase the CR Yamabe constant at later time.
To be precise, 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 (see, e.g. [Lee, Web]). The characteristic vector …eld of is the unique vector …eld T such that (T ) = 1 and d (T; ) = 0. A CR structure compatible with is a smooth endomorphism J : ! such that J2 = identity.
A pseudohermitian structure is a CR structure J compatible with together with a global contact form .
Given a pseudohermitian structure (J; ), we can choose a complex vector …eld Z1,
an eigenvector of J with eigenvalue i, and a complex 1-form 1 such that f ; 1; 1g is dual to fT; Z1; Z1g. It follows that d = ih11 1^ 1 for some nonzero real function
h11. If h11 is positive, we call such a pseudohermitian structure (J; ) positive, and
we can choose a Z1 (hence 1) such that h11= 1. That is to say
d = i 1^ 1:
1991 Mathematics Subject Classi…cation. 53C21, 32G07.
Key words and phrases. CR Yamabe constant, Cartan ‡ow, Spherical CR structure.
We will always assume our pseudohermitian structure (J; ) is positive and h11= 1
throughout the paper. The pseudohermitian connection of (J; ) is the connection r on T M C (and extended to tensors) given by
rZ1 = !11 Z1; rZ1 = !11 Z1; rT = 0;
in which the 1-form !11 is uniquely determined by the following equation and
asso-ciated normalization condition:
d 1 = 1^!11+ A11 ^ 1; !11+ !11 = 0:
The coe¢ cient A1
1 is called the (pseudohermitian) torsion. Since h11 = 1, A11 =
h11A11 = A11. And A11 is just the complex conjugate of A11. Di¤erentiating !11
gives
d!11 = R 1^ 1+ 2iIm(A11;1 1^ )
where R is the Tanaka-Webster curvature (see [Tan, Web]).
We can de…ne the covariant di¤erentiations with respect to the pseudohermitian connection. For instance, f;1 = Z1f ; f;11 = Z1Z1f !11(Z1)Z1f; and f;0 = T f for
a (smooth) function f . We de…ne the subgradient operator rb and the sublaplacian
operator b by
rbf = f;1Z1+ f;1Z1; bf = f;11 f;11;
respectively. Also we de…ne jrbfj 2
J; = 2f;1f;1 for real f:
Recall that the CR Yamabe constant on a closed CR 3-manifold is de…ned by
(0.1) Y(J) + inf u2C1(M );u>0 R M 4jrbuj 2 J; + Ru2 ^ d R Mu 4 ^ d 1=2 ;
where R is the Tanaka-Webster curvature and ^ d is the volume form associated with (J; ). The Euler-Lagrangian equation for a minimizer u is
4 bu + Ru = Y(J)u3; (0.2) Z M u4 ^ d = 1: (0.3)
Note that the existence of minimizer u follows from the solution of CR Yamabe problem (e.g., see the serial papers [JL1, JL2, JL3]). Given a such solution u the
contact form u2
has constant Tanaka-Webster curvature Y(J): Like the Riemannian case, we de…ne the sigma invariant of M by
(M ) = sup
J Y(J);
where the sup is taken over all pseudohermitian structures (J; ) on M .
We also recall that the Cartan ‡ow is deforming the CR structure in the direction of its Cartan tensor. Due to a result of Gray [Gra], we lose no generality by …xing a contact bundle and contact form throughout this paper. This gives rise to the evolution equation
(0.4) @
@tJ(t) = 2QJ(t);
where QJ = 2 Re(iQ11 1 Z_1) is the Cartan tensor of the CR structure J with
Q11= 1 6R;11+ i 2RA11 A11;0 2i 3A11; _ 11
(Lemma 2.2 in [CL]), whose vanishing characterizes the spherical J : QJ = 0 if and
only if J is spherical.
We remark that (0.4) is a fourth order nonlinear subparabolic equation which is the negative gradient ‡ow of the Burns-Epstein invariant and the spherical CR structures are the only equilibrium solutions . The short time existence of solutions for (0.4) is proved by adding a gauge-…xing term to the right-hand side of (0.4) (see Theorem B in [CL]).
Now we can state our …rst result.
Proposition 0.1. Let J(t); t2 [0; ) for some > 0; be a solution of the Cartan ‡ow
(0.4) on a closed 3-dimensional pseudohermitian manifold M . Assume that there is a C1-family of smooth functions u(t) > 0; t2 [0; ) which satisfy
4 bu(t)+ RJ (t)u(t) = Ye(t)u 3 (t); (0.5) Z M u4(t) ^ d = 1; (0.6)
where eY is function of t only. Then we have d dtYe(t) = Z M 2 Re[8Q11u;_1u;_1 + 2 3A _ 1 _ 1(A11;1u;1+ A11;1u;1)u] ^ d + Z M 1 3[Re(iA11R; _ 1 _ 1) + 2jrbA11j 2 J; + 7RjA11j2J; ]u 2 ^ d (0.7) Z M 2 Re(Q11;_ 1 _ 1 + 4 3iA11A _ 1 _ 1;0)u 2 ^ d ;
where u = u(t) and Q11; A11; R; and rb are the Cartan tensor, the torsion, the
Tanaka-Webster curvature, and the subgradient of (J(t); ); respectively.
Proof. Let be a …xed contact form and let dudt + h: Note that e
Y(t)=
Z
M
4jrbuj2J; + Ru2 ^ d :
By applying the computation on page 231-232 in [CL] (with E11 replaced by iQ11),
we compute d dtYe(t) = Z M [16 Re(Q11u;_1u;_1) + 8u bh] ^ d + Z M [ 2 Re(Q11;_ 1 _ 1 + iA _ 1 _ 1Q11)u 2+ 2Rhu] ^ d ; (0.8)
where we have used
@ @t jrbuj 2 J; = 4 Re(Q11u; _ 1u; _ 1 + h;1u; _ 1) and @ @tR = 2 Re(Q11; _ 1 _ 1 + iA _ 1 _ 1Q11):
Taking derivative d=dt of (0.5), we obtain
4 bh 16 Re(Q11u;_1_1 + Q11;_1u;_1) 2 Re(Q11;_1_1 + iA_1_1Q11)u + Rh
= [d
dtYe(t)]u
3+ 3 e
Y(t)hu2:
Multiplying this by 2u; we get
8u bh + 2Rhu = 32Re(Q11u;_1_1u + Q11;_1u;_1u) +4 Re(Q11;_ 1 _ 1 + iA _ 1 _ 1Q11)u 2 +[d dtYe(t)]2u 4 + 6 e Y(t)hu3:
Substituting this into (0.8) to eliminate h; we obtain d dtYe(t) = Z M 16 Re(Q11u;_1u;_1 + 2Q11u;_1_1u + 2Q11;_1u;_1u) ^ d + Z M 2 Re(Q11;_ 1 _ 1 + iA _ 1 _ 1Q11)u 2 ^ d + 2dtdYe(t) +6 eY(t) Z M hu3 ^ d : Integrating by parts, we have
(0.9) d dtYe(t) = Z M [16 Re(Q11u;_1u;_1) 2 Re(Q11;_1_1 + iA_1_1Q11)u2] ^ d ; where we used Z M (Q11u;_1u;_1 + Q11u;_1_1u + Q11;_1u;_1u) ^ d = 0 and Z M hu3 ^ d = 0 which is obtained by taking derivative d=dt of (0.6).
Next by using the commutation relation for pseudohermitian covariant derivative
(0.10) A 11;1 _ 1 A11; _ 11= iA11;0+ 2A11R;
the Cartan tensor Q11 can also be represented by
Q11 = 1 6[R;11+ 7iRA11 8A11;0 2i(A11; _ 11+ A11;1 _ 1)]: Hence we have 2 Re(iA_ 1 _ 1Q11) = 1 3Re[ iA _ 1 _ 1R;11+ 7RjA11j 2 J; + 8iA_1_1A11;0 2A_1_1(A11;_11+ A11;1_1)]:
The proposition follows from integrating by parts to the last term of above equation into (0.9).
We say a function f is basic if T f = 0. Then we have the following Corollary. Corollary 0.2. Let (M; J0; ) be a closed 3-dimensional pseudohermitian manifold of
nonnegative constant Tanaka-Webster curvature and the real part (or imagine part) of torsion is basic. Let J(t) be the solution of the Cartan ‡ow (0.4) with J(0) = J0
and assume that there is a C1-family of smooth functions u
> 0 satisfy the assumption in Proposition 0.1. Then we have d
dtjt=0Ye(t) 0 and the
equality holds if and only if J0 is spherical.
Proof. Integrating by parts, we obtain Z M Re(Q11;_ 1 _ 1 + 4 3iA11A _ 1 _ 1;0) ^ d = 0;
which follows from the fact that Re(A11)(or Im(A11)) is basic. Now since rbu(0) = 0
and R 0 be a constant, we have
d dtjt=0Ye(t) = 1 3[u(0)] 2 Z M [2jrbA11j2J; + 7RjA11j2J; ] ^ d 0:
Suppose the above equality holds. If R is positive, then A11 = 0. If R is zero, then
by the fact rbA11 = 0 and the commutation relation (0.10) will imply A11;0= 0. All
these implies that Q11= 0 and therefore J is spherical. Conversely, if J is spherical,
that is Q11= 0: Then from (0.9), we get
d
dtjt=0Ye(t) =
Z
M
[16 Re(Q11u;_1u;_1) 2 Re(Q11;_1_1 + iA_1_1Q11)u2] ^ d = 0:
This implies the Corollary.
Note that eY(t) in Corollary 0.2 may not equal to the CR Yamabe constant Y(J(t))
even if J(0) satis…es eY(0) =Y(J(0)):If we assume that the real part (or imagine part) of
torsion of (J(0); )is basic and (J(t); ) has unit volume and constant Tanaka-Webster
curvature Y(J(t)); we have the following result, which says that in…nitesimally the
Cartan ‡ow will try to increase the CR Yamabe constant.
Corollary 0.3. (i) Let (M; J0; ) be a closed 3-dimensional pseudohermitian manifold
of nonnegative constant Tanaka-Webster curvature and the real part (or imagine part) of torsion is basic. Let J(t) be the solution of the Cartan ‡ow (0.4) with J(0) = J0 and
assume that there is a C1-family of smooth functions u(t) > 0; t2 [0; ) for some > 0
with constant u(0) such that (J(t); ) has unit volume and constant Tanaka-Webster
curvature Y(J(t)): Then we have dtdjt=0Y(J(t)) 0 and the equality holds if and only
if J0 is spherical.
Acknowledgments. The author would like to express his thanks to Prof. S.-C. Chang for bring his attention on the paper [CLu] and Prof. P. Lu for valuable discussions. The author’s research is partially supported by NSC of Taiwan.
References
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[Gra] J. W. Gray, Some global properties of contact structures, Ann. Math. 69 (1959), 421-450. [JL1] D. Jerison and J. M. Lee, The Yamabe problem on CR manifolds, J. Di¤. Geom., 25 (1987),
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[Web] S. M. Webster, Pseudohermitian structures on a real hypersurface, J. Di¤. Geom. 13 (1978), 25-41
Department of Applied Mathematics, National PingTung University of Education, PingTung 90003, Taiwan, R.O.C
