ON CR PERELMAN’S HARNACK ESTIMATES AND ENTROPY
FORMULAE FOR THE COUPLED CR YAMABE FLOW
SHU-CHENG CHANG
In this note, we …rst derive the CR analogue of Perelman’s Harnack estimate (0.4)
for the positive solution of the CR conjugate heat equation
@u
=
@t
(0.1)
4
bu
+ 4W u:
under the CR Yamabe ‡ow. Then we de…ne the CR Perelman’s F-functional (0.5)
from this Perelman’s Harnack quantity and obtain its monotonicity property for the
following coupled CR Yamabe ‡ow
8
< @ (t) = 2W (t) (t) ;
@t
(0.2)
: @f = 4 b f + 4 jrb f j2
@t
(0) = ;
4W:
Second, we derive the CR analogue of Perelman’s Harnack estimate (0.6) for the
positive fundamental solution of the CR conjugate heat equation (0.1) under the CR
Yamabe ‡ow. Then we de…ne the CR Perelman’s W-functional (0.8) and obtain its
monotonicity property for the following coupled CR Yamabe ‡ow
(0.3)
8
>
>
>
<
>
>
>
:
@
@t
@f
@t
=
4
d
dt
=
1:
(t) =
2W (t) (t) ;
bf
+ 4 jrb f j2
(0) = ;
4W + 2 ;
Now we state the …rst Perelman’s Li-Yau type Harnack estimate for the positive
solution of the CR conjugate heat equation under the CR Yamabe ‡ow in a closed
spherical pseudohermitian 3-manifold with nonnegative Tanaka-Webster curvature
and vanishing torsion.
Key words and phrases. CR heat equation, CR Yamabe soliton, Li-Yau gradient estimate, Perelman Harnack estimate, Conjugate heat equation, Entropy formulae, Spherical structure, Coupled
CR Yamabe ‡ow.
1
2
SHU-CHENG CHANG
Theorem 0.1. Let (M; J; ) be a closed spherical pseudohermitian 3-manifold with
nonnegative Tanaka-Webster curvature and vanishing torsion. Let (M; J; (t)) be a
solution to the CR Yamabe ‡ow on M
[0; T ] and u be the positive solution to the
CR conjugate heat equation (0.1) with u = e
(0.4)
on M
bf
[0; T ) with
=T
f
and f0 (x; 0) = 0: Then
1
3
jrb f j2 + W
4
2
1
0
t:
As pointed out by S.-T. Yau that the derivation of the entropy formula resembles
Li-Yau gradient estimate for the heat equation. We de…ne the CR Perelman Ffunctional by
R
F( (t); f (t)) = 4 M [( b f 34 jrb f j2 + 21 W )]e
(0.5)
R
= M (2W + jrb f j2 )e f d :
R
with the constraint M e f d = 1:
f
d
We derive the following monotonicity property of CR F-functional.
Theorem 0.2. Let (M; J; ) be a closed spherical pseudohermitian 3-manifold with
nonnegative Tanaka-Webster curvature and vanishing torsion. Let (M; (t); f (t)) be
a solution to the coupled system (0.2) on M
d
F(
dt
(t); f (t)) = 8
0
R
M
rH
2
f+
[0; T ) with f0 (x; 0) = 0. Then
W
L
2
2
ud + 2
R
M
W jrb f j2 ud
for all t 2 [0; T ):
Next we state the second CR Perelman’s Li-Yau type Harnack estimate for the positive fundamental solution of the CR conjugate heat equation under the CR Yamabe
‡ow.
Theorem 0.3. Let (M; J; ) be a closed spherical pseudohermitian 3-manifold with
positive Tanaka-Webster curvature and vanishing torsion. Let (M; J; (t)) be a solution to the CR Yamabe ‡ow on M
[0; T ] and u be the positive fundamental solution
to the CR conjugate heat equation (0.1) with u =
(0.6)
bf
e
(4
3
1
1
jrb f j2 + W + (f
4
2
8
f
)2
and f0 (x; 0) = 0: Then
4)
0:
CR PERELMAN’S HARNACK ESTIMATES AND ENTROPY FORMULAE
3
That is
f + jrb f j2
(0.7)
on M
[0; T ) with
=T
2W +
f
2
0
t:
Again we de…ne the CR Perelman W-functional by
(0.8)
R
W( (t); f (t); ) = 4 M [ b f 34 jrb f j2 + 21 W + 81 (f 4)](4 ) 2 e
R
= M [ (2W + jrb f j2 ) + 12 (f 4)](4 ) 2 e f d ;
with the constraint
7 ! c
and
di¤eomorphism
R
M
(4
) 2e
f
f
d
= 1: Note that W is scalar invariant under
d
7 ! c : Moreover, W( ; f; ) = W(
;f
; ) for a contact
: M ! M:
We will derive the following monotonicity property of CR W-functional.
Theorem 0.4. Let (M; J; ) be a closed spherical pseudohermitian 3-manifold with
nonnegative Tanaka-Webster curvature and vanishing torsion. Let (M; (t); f (t); (t))
be a solution to the coupled CR Yamabe ‡ow on M
d
dt W(
=8
+
0
[0; T ) with f0 (x; 0) = 0. Then
(t); f (t); (t))
R
M
R
M
rH
2
f+
W
L
2
[2W jrb f j2 +
jrb f j
1
4
2
2
L
ud
]ud
for all t 2 [0; T ):
The classi…cation of CR Yamabe soliton is one of the key steps in understanding
the singularity formation of the Yamabe ‡ow.
Theorem 0.5. (i) Any closed spherical 3-dimensional steady CR Yamabe breather
with nonnegative Tanaka-Webster curvature and vanishing torsion is a closed pseudohermitian 3-manifold of zero Tanaka-Webster curvature and vanishing torsion.
(ii) Any closed spherical 3-dimensional shrinking CR Yamabe breather with nonnegative Tanaka-Webster curvature and vanishing torsion is a closed pseudohermitian
3-manifold of positive constant Tanaka-Webster curvature and vanishing torsion.
4
SHU-CHENG CHANG
Finally, if we further assume that our solution to the CR Yamabe ‡ow is of Type
I. That is
jW j
k0
;
T t
for some constant k0 . Here T is the blow-up time. Thus we have the Perelman’s LiYau type Harnack estimate for the positive solution of the CR conjugate heat equation
under the CR Yamabe ‡ow in a closed spherical pseudohermitian 3-manifold with
vanishing torsion.
Theorem 0.6. Let (M; J; ) be a closed spherical pseudohermitian 3-manifold with
vanishing torsion. Let (M; J; (t)) be a solution of Type I to the CR Yamabe ‡ow
(??) on M
[0; T ) and u be the positive solution to the CR conjugate heat equation
(0.1) with u = e
f
and f0 (x; 0) = 0: Then, for some
bf
on M
[0; T ) with
=T
1
3
jrb f j2 + W
4
2
0
0
=
0 (k0 )
>0
0
t:
Acknowledgments. The …rst author would like to express his thanks to Prof.
S.-T. Yau for his inspirations from Li-Yau gradient estimate, Prof. C.-S. Lin, director
of Taida Institute for Mathematical Sciences, NTU, for constant encouragement and
supports. The work is not possible without their e¤orts.
Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan,
Taida Institute for Mathematical Sciences (TIMS), National Taiwan University,
Taipei 10617, Taiwan
E-mail address: [email protected]