柯西黎曼 Li-Yau-Hamilton 不等式即其應用

國立臺灣大學理學院數學所 博士論文

Department of Mathematics College of Science

National Taiwan University Doctoral Dissertation

柯西黎曼 Li-Yau-Hamilton 不等式及其應用

CR Li-Yau-Hamilton Inequality and its Applications

樊彥彣 Yen-Wen Fan

指導教授：張樹城 博士 Advisor: Shu-Cheng Chang , Ph.D.

中華民國 103 年 6 月 June 2014

iii

誌 謝

好的建議。同時感謝鄭日新老師在我當研究助理過程中不嗇惜給予許多協助。也感謝陳瑞堂 老師和吳進通老師的關心和許多學長學弟的熱心討論；如郭庭榕對論文的修改建議和文章的 寫法有很好的直覺，常提醒一些需要注意的地方，也常與我討論問題。張覺心對方程的見解 對我在論文上也很有幫助，讓我在討論中學習了許多。還有許多位朋友，感謝你們陪伴我學 習的過程。

iv

摘 要

這篇文章包含三大部分，第一部分證明矩陣形式的 Li-Yau-Hamilton Harnack 不等式。第 二部份延續第一部分的工作，推廣至(1,1)-form 形式的 Li-Yau-Hamilton Harnack 不等式。第三 部份將應用這不等式証明柯西黎曼上的 Gap 定理。

關鍵字：擬埃爾米特，Li-Yau-Hamilton，Gap 定理，Harnack 不等式

In the …rst part of thesis, we …rst derive the CR analogue of matrix Li-Yau-Hamilton inequality for a positive solution to the CR heat equation in a closed pseudohermitian (2n+1)- manifold with nonnegative bisectional curvature and bitorsional tensor. We then obtain the CR Li-Yau gradient estimate in a standard Heisenberg group. Finally, we extend the CR matrix Li-Yau-Hamilton inequality to the case of Heisenberg groups. As a consequence, we derive the Hessian comparison property in the standard Heisenberg group.

In the second part, we study the CR Lichnerowicz-Laplacian heat equation deformation of (1; 1)-tensors on a complete strictly pseudoconvex CR (2n+1)-manifold and derive the linear trace version of Li-Yau-Hamilton inequality for positive solutions of the CR Lichnerowicz- Laplacian heat equation. We also obtain a nonlinear version of Li-Yau-Hamilton inequality for the CR Lichnerowicz-Laplacian heat equation coupled with the CR Yamabe ‡ow and trace Harnack inequality for the CR Yamabe ‡ow.

In the last part, by applying a linear trace Li-Yau-Hamilton inequality for a positive (1; 1)-form solution of the CR Hodge-Laplace heat equation and monotonicity of the heat equation deformation, we obtain an optimal gap theorem for a complete strictly pseudocovex CR (2n + 1)-manifold with nonnegative pseudohermitian bisectional curvature and vanishing torsion. We prove that if the average of the Tanaka-Webster scalar curvature over a ball of radius r centered at some point o decays as o (r ²), then the manifold is ‡at.

v

1. Abstract v

2. Introduction 1

2.1. CR Li-Yau Gradient Estimate and Harnack Inequality 2

2.2. CR Matrix Li-Yau-Hamilton Inequality 4

2.3. CR Linear Trace Li-Yau-Hamilton Inequality and Gap Theorem 6

2.4. The Coupled CR Yamabe Flow 7

3. Preliminary 10

4. CR Matrix Li-Yau-Hamilton Harnack Inequality 12

4.1. CR Matrix Li-Yau-Hamilton Inequality 15

4.2. The CR Gradient Estimate and Harnack inequality in Heisenberg Groups 20

4.3. Complete noncompact case 25

5. Linear Trace Li-Yau-Hamilton inequality 31

5.1. The CR Bochner-Weitzenbock Formula 35

5.2. Linear Trace Li-Yau-Hamilton Inequality 39

5.3. Nonlinear Version for Li-Yau-Hamilton Inequality 50

6. CR Gap Theorem 58

6.1. CR Moment-Type Estimates 59

6.2. CR Lichnerowicz-Laplacian heat equation 63

6.3. Proof of CR Optimal Gap Theorem 67

Appendix A. 71

References 74

1

2. Introduction

We brie‡y introduce our works and results.

In the seminal paper [LY], P. Li and S.-T. Yau established the parabolic Li-Yau Harnack estimate for the positive solution u(x; t) of the time-independent heat equation

(2.1) @u (x; t)

@t = u (x; t)

in a complete Riemannian l-manifold with nonnegative Ricci curvature. Here is the Laplace-Beltrami operator. Later in [H2], Richard Hamilton extended the Li-Yau estimate to the full matrix version of the Hessian estimate of u under the stronger assumptions that M is Ricci parallel and of nonnegative sectional curvature. Furthermore, Hamilton ([H1]) proved the matrix Harnack inequality for solutions to the Ricci ‡ow

(2.2) @g_ij(x; t)

@t = 2R_ij(x; t)

when the curvature operator is nonnegative. This inequality is called the “Li-Yau-Hamilton”

type estimates. Since then, there are many additional works in this direction which cover various di¤erent geometric evolution equations such as the mean curvature ‡ow ([H2]), the Kähler-Ricci ‡ow ([Ca]), the Yamabe ‡ow ([C]), etc.

On the other hand, the Kähler-Ricci curvature (1; 1)-tensor of a Kähler-Ricci ‡ow solu- tion satis…es a Lichnerowicz-Laplacian heat equation. In general, the Hodge-Laplacian heat equation on symmetric (p; p)-tensors is a geometrically interesting system and has been exten- sively studied since the original works of Hodge and Kodaira ( [Mo] and references therein).

For instances, we refer to the Lichnerowicz-Laplacian heat equation on (1; 1)-tensors and the Hodge-Laplacian heat equation on (p; p)-tensors as in [NN].

Along this line with method of Li-Yau gradient estimate, H.-D. Cao and S.-T. Yau ([CY]) studied the heat equation

(2.3) @u (x; t)

@t = Lu(x; t)

in a closed l-manifold with a positive measure and a subelliptic operator with respect to the sum of squares of vector …elds L = Ph

i=1X_i² Y; h l with Y = Ph

i=1c_iX_i where X₁; X₂; :::; X_h are smooth vector …elds satisfying Hörmander’s condition : the vector …elds together with their commutators up to …nite order span the tangent space at every point of M: Suppose that [Xi; [X_j; X_k]] can be expressed as linear combinations of X1; X₂; :::; X_h and their brackets [X1; X₂]; :::; [X_{l 1}; X_h]: They showed that the gradient estimate for the positive solution u(x; t) of (2.3) on M [0;1):

In the …rst part of this paper, we focus on the CR Li-Yau-Hamilton type gradient estimate for the positive solution u(x; t) of the CR heat equation

(2.4) @u (x; t)

@t = bu (x; t) :

and for the positive symmetric (1; 1)-form (x; t) of CR Lichnerowicz-Laplacian heat equation

(2.5) @

@t = 4 _b + 2R (R + R ) :

On the other hand, we are also interested in the coupled heat equation. Let (t) be a family of smooth contact forms and J (t) be a family of CR structures on (M³; J₀; ) with J (0) = J0

and (0) = . In the paper of [CKW], we consider the following torsion ‡ow which is the CR analogue of the Hamilton Ricci ‡ow:

(2.6)

@

@tJ = 2A_J; ;

@

@t = 2R :

on M [0; T ) with J (t) = i ¹ Z₁ i ¹ Z₁ and AJ; (t) = A₁₁ ¹ Z₁ + A₁₁ ¹ Z₁: In particular if the initial torsion is vanishing, the torsion ‡ow (2.6) is equivalent to the CR Yamabe ‡ow

(2.7)

@

@t (t) = 2R (t) (t) ; (0) = ;

in a closed CR 3-manifold. We will study the Li-Yau-Hamilton inequality for the coupled CR Yamabe ‡ow as well.

Finally, we recall some de…nitions as followings.

De…nition 2.1. Let (M; J; ) be a closed pseudohermitian 3-manifold. We call a CR struc- ture J spherical if Cartan curvature tensor Q11 vanishes identically. Here

Q₁₁ = 1

6W₁₁+ i

2W A₁₁ A_11;0 2i

3A_11;^_₁₁:

Note that (M; J; ) is called a closed spherical pseudohermitian 3-manifold if J is a spherical structure. We observe that the spherical structure is CR invariant and a closed spherical pseudohermitian 3-manifold (M; J; ) is locally CR equivalent to (S³; bJ ; b):

De…nition 2.2. Let (M; J; ) be a closed pseudohermitian (2n + 1)-manifold with = ker . A piecewise smooth curve : [0; 1] ! M is said to be a Legendrian curve if _ ( ) 2 whenever _ ( ) exists. The length of is then de…ned by

l( ) = Z 1

0

(h _ ( ); _ ( ) iL )¹²d :

The Carnot-Carathéodory distance dcc between two points p; q 2 M is de…ned by d_cc(p; q) = inffl( )j 2 C^p;qg ;

where Cp;q is the set of all Legendrian curves which join p and q.

2.1. CR Li-Yau Gradient Estimate and Harnack Inequality. Let u be the positive solution of (2.4) and denote

f (x; t) = ln u (x; t) : Then f (x; t) satis…es the equation

(2.8) b

@

@t f (x; t) = jr^bf (x; t)j²:

We observe that one of di¢ culties is to deal with CR Bochner formula which involving a term hJr^bf; r^bf₀i that has no analogue in the Riemannian case. In order to overcome this di¢ culty, we introduce a new scalar Harnack quantity

(2.9) F (x; t; a; c) = t jr^bfj²(x) + af_t+ ctf₀²(x) ;

with f = ln u by adding an extra term tf₀² to jr^bfj² + af_t which was appeared in Li-Yau estimate ([LY]). Then one can derive CR versions of Li-Yau gradient estimates and classical Harnack inequality.

Theorem 2.1. ([CKL1]) Let (M; J; ) be a closed pseudohermitian (2n + 1)-manifold.

Suppose that

2Ric (X; X) (n 2)T or (X; X) 0 for all X 2 T^1;0 T0;1: If u (x; t) is the positive solution of

b

@

@t u (x; t) = 0 with

[ _b; T] u = 0

on M [0; 1) : Then f (x; t) = ln u (x; t) satis…es the following subgradient estimate

(2.10) jr^bfj² (1 + 3

n)f_t+n

3t(f₀)² < (⁹_n+ 6 + n)

t :

When the manifold is complete noncompact, the proof of CR Li-Yau gradient estimate (2.11) relies on the CR sub-Laplacian comparison property and the extra u0-growth property with ju⁰j ^Ctuthat has no analogue in the Riemannian case. However, both properties holds in a standard Heisenberg group Hⁿ which is ‡at and vanishing torsion. Then we are able to derive the following CR Li-Yau gradient estimate on Hⁿ:

Theorem 2.2. ([CFTW]) Let (Hⁿ; J; ) be the standard (2n + 1)-dimensional Heisenberg group. If u(x; t) is the positive solution of the CR heat equation (2.4) on Hⁿ [0;1). Let ' = ln u; for any < 1; then there exists a positive constant C depending on such that (2.11) jr^b'j²+ '_t+ t'²₀ ^C_t:

By applying Theorem 2.2, we have the following CR Liouville-type theorem for a positive pseudoharmonic function u on (Hⁿ; J; ) which recaptured the Liouville theorem due to Chang-Kuo-Tie [CKT] and Koranyi and Stanton ([KS]) by a di¤erent method.

Corollary 2.3. Let (Hⁿ; J; ) be the standard (2n + 1)-dimensional Heisenberg group. If u(x; t) is the positive smooth function with bu = 0; then u(x; t) is constant. That is, there does not exist any positive nonconstant pseudoharmonic function in Hⁿ.

By using the method of CR Li-Yau gradient estimate ([LY], [CKL1]) and CR Bochner formula, we derive a CR gradient estimate and CR Harnack inequality for the positive solution of the CR heat equation (2.4) in (2n + 1)-dimensional Heisenberg group.

Corollary 2.4. Let (Hⁿ; J; ) be the standard (2n + 1)-dimensional Heisenberg group. If u(x; t) is the positive solution of the CR heat equation (2.4) on Hⁿ [0;1) ; we have the Harnack inequality

(2.12) ^u(x_u(x^1;t1)

2;t2) t2

t1

C

exp ^d_2(t^c(x1;x2)2

2 t1)

for any x1; x₂ in Hⁿ and 0 < t1 < t₂ < 1; where d^c(x₁; x₂) is the Carnot-Carathéodory distance between x1 and x2.

As a consequence of Corollary 2.4 and [CY], we have the following upper bound estimate for the heat kernel of (2.4).

Corollary 2.5. Let (Hⁿ; J; ) be the standard (2n + 1)-dimensional Heisenberg group and H(x; y; t) be the heat kernel of (2.4) on M [0;1). Then for some constant > 1 and 0 < < 1; H(x; y; t) satis…es the estimate

(2.13) H(x; y; t) C( ) V ¹²(B_x(p

t))V ¹²(B_y(p

t)) exp d²_cc(x; y) (4 + )t with C( ) ! 1 as ! 0.

Once we have the upper bound estimate for the heat kernel and the sub-laplacian com- parison property, then by applying the arguments of Li-Tam as in [LT] or [Li], we have the following mean value inequality.

Corollary 2.6. Let (Hⁿ; J; ) be the standard (2n + 1)-dimensional Heisenberg group and g be subsolution of the CR heat equation such that

@

@t ^b g (x; t) 0:

Then for some constant C depend on ; ; , such that 0 < < 1; 0 < < T, 0 < < ¹₂; the following inequality holds for any > 2p

T ;

(2.14) sup

Bp((1 ) ) [ ;T ]

g C

Z T (1 )

Z

Bp( )

g (y; s) dyds:

2.2. CR Matrix Li-Yau-Hamilton Inequality. Let u(x; t) be the positive solution of the CR heat equation (2.4). For the CR Li-Yau gradient estimate as in the paper [CKL1], we observe that one of di¢ culties is to deal with CR Bochner formula which involving a term hJr^bf; r^bf₀i that has no analogue in the Riemannian case. In order to overcome this di¢ culty, we introduce a new scalar Harnack quantity F = t[jr^bfj²+ f_t+tf₀²]with f = ln u by adding an extra term tf₀² to jr^bfj²+ f_t which was appeared in Li-Yau estimate ([LY]).

Now we want to …nd the right quantity for the CR matrix Li-Yau-Hamilton inequality.

By comparing the Harnack quantity in [CN] in the case of Kähler manifolds, we de…ne the matrix Harnack quantity

(2.15) N = 1

2(u + u ) + 2u

th bu u

u atju⁰j² u h

by adding an extra term F := at^ju_u^0j2h in which positive constants a and b to be deter- mined later (say a = ₂₄¹and b = ¹₄).

De…nition 2.3. ([GL]) De…ne the purely holomorphic Hessian operator P : P ' := 2i(A ' )

and the purely holomorphic Poisson operator Q :

Q' := h (P ') = 2i(A ' )

for any smooth function ': Note that P ' = 0 = Q' for any smooth function ' if A = 0 on M:

Then, based on the following key estimate, we have Theorem 2.7.

Lemma 2.1. Let u(x; t) be the positive solution of the CR heat equation (2.4). Then

1

2(u + u ) satis…es the following : 1

2

@

@t ^b u + u = 2R u R u R u + C ;

where

C := i A _; u A _; u h + i A u A u h

+in A u A u + in A _; u A _; u

:= (Re Qu)h + n(Re P u):

Note that trC = h C = 0: In particular we have C₁₁= 0 for n = 1: In addition if the positive solution u satis…es P u = 0 which is the case when the torsion is vanishing, then u satis…es the following CR Lichnerowicz-Laplacian heat equation ([CCF]) :

@

@t ^b u = 2R u R u R u :

Hence we have the following CR analogue of matrix Li-Yau-Hamilton inequality for any positive solution u to (2.4).

Theorem 2.7. ([CFTW]) Let M be a closed pseudohermitian (2n + 1)-manifold with non- negative bisectional curvature and nonnegative bi-torsional tensor. Let u be the positive solution of the CR heat equation (2.4). In addition if the positive solution u satis…es the purely holomorphic Hessian operator P u = 0: Then

(2.16) (u + u ) + 1

2[(u V + u V ) + uV V ] t 12

ju⁰j²

u h +4

tuh 0

for t > 0 and any vector …led V = V of type (1; 0) on M: Here P is the purely holomorphic Hessian operator (De…nition 2.3). In particular, the CR matrix Li-Yau-Hamilton inequality (2.16) holds in a closed pseudohermitian (2n + 1)-manifold of nonnegative bisectional cur- vature and vanishing torsion. If we choose the optimal V = ru=u and take the trace of (2.16), we recapture the CR Li-Yau gradient estimate (2.10).

When the manifold is complete noncompact, we will need to use the CR Li-Yau Harnack inequality (2.12) and Li-Tam mean value inequality (2.14) in the proof of the CR matrix Li-Yau-Hamilton inequality (2.16). However, both estimates hold in a standard Heisenberg group Hⁿ which is ‡at and vanishing torsion. Then as a consequence of Theorem 2.7, we are able to derive the following CR matrix Li-Yau-Hamilton inequality on Hⁿ:

Theorem 2.8. ([CFTW]) Let (Hⁿ; J; ) be the standard (2n + 1)-dimensional Heisenberg group. If u(x; t) is the positive solution of the CR heat equation (2.4) on Hⁿ [0;1). Then the CR matrix Li-Yau-Hamilton inequality (2.16) holds.

By applying Theorem 2.8 to the heat kernel H(x; y; t) with V = ^rH_H , we have the following complex Hessian comparison theorem for r on Hⁿ. Such a Hessian comparison property seems to be new in the standard (2n + 1)-dimensional Heisenberg group Hⁿ: Corollary 2.9. Let (Hⁿ; J; ) be the standard (2n + 1)-dimensional Heisenberg group. Then in the sense of distribution, we have

[(r²(x)) + (r²(x)) ] (16 + C₀)h (x)

for some constant C0. In particular, we recapture the sub-Laplacian comparison property

br²(x) (16 + C₀)n in the Heisenberg group.

2.3. CR Linear Trace Li-Yau-Hamilton Inequality and Gap Theorem. We now consider the CR Hodge-Laplacian

H = 1

2( _b + _b)

for Kohn-Rossi Laplacian b. For any (1; 1)-form (x; t) = ^ ; we study the CR Hodge-Laplacian heat equation on M [0; T )

@

@t (x; t) = 4 _H (x; t)

in which connects to the existence problem of pseudo-Einstein CR (2n + 1)-manifolds with n 2. It follows from the CR Bochner-Weitzenbock Formula that the CR parabolic equation above is equivalent to the CR analogue of Lichnerowicz-Laplacian heat equation (2.5).

De…ne the Harnack quadratic by (2.17)

Z (x; t) (V ) := k₁ 1

2 (div ) _; + (div ) _; + (div ) V + (div ) V + V V +H t for any vector …eld V 2 T^1;0(M ) ; H = h and k1 to be determined later. Moreover,

;0 is denoted the component of covariant derivative of the tensor with Reeb vector …eld T:

The following is linear trace Li-Yau-Hamilton inequality for the CR Lichnerowicz-Laplacian heat equation.

Theorem 2.10. ([CCF]) Let (M; J; ) be a closed strictly pseudoconvex CR (2n+1)-manifold with nonnegative bisectional curvature and vanishing torsion. Let (x; t) be a nonnegative symmetric (1; 1)-tensor satisfying the CR Lichnerowicz-Laplacian heat equation (2.5) on M [0; T ) and _;0(x; 0) = 0 at t = 0. In additional if M is complete noncompact, we assume that there exists a constant a > 0 such that

Z T Z

M

e ^ar2k (x; t)k²d dt <1

and Z T Z

M

e ^ar2kr^T (x; t)k²d dt <1;

where r (x) is the Carnot-Carathéodory distance from a …xed point o and any > 0. Then Z (x; t) 0;

for 0 < k1 8:

Then, based on the linear trace Li-Yau-Hamilton inequality for the CR Lichnerowicz- Laplacian heat equation. Lichnerowicz-Laplacian heat equation and then CR monotonicity of heat equation deformation of positive (1; 1)-forms, we have the following CR gap Theorem.

Theorem 2.11. ([CF]) Let M be a complete noncompact strictly pseudoconvex CR (2n + 1)- manifold with nonnegative bisectional curvature and vanishing torsion. Then M is ‡at if

(2.18) 1

V_o(r) Z

Bo(r)

R (y) d (y) = o r ² ;

for some point o 2 M: Here R (y) is the Tanaka-Webster scalar curvature and V^o(r) is the volume of the ball Bo(r) with respect to the Carnot-Carathéodory distance. As a consequence if M is not ‡at, then

lim inf

r !1

r² V_o(r)

Z

Bo(r)

R (y) d (y) > 0 for any o 2 M:

2.4. The Coupled CR Yamabe Flow. We …rst study the following time-dependent CR heat equations with potentials

(2.19) @u

@t = 4 _bu cRu

evolving by the CR Yamabe ‡ow on M [0; T ). Here b is the time-depending sublaplacian and R(t) is the Tanaka-Webster scalar curvature with respect to the contact form (t). We will derive di¤erential Harnack estimates for positive solutions of (2.19) for c = 2:

We also present its application of Theorem 2.7 to obtain the nonlinear version of Harnack inequality for CR Lichnerowicz-Laplacian heat equation (2.5) coupled with the CR Yamabe

‡ow (5.15).

We expect our Harnack estimate will play an important role in the study of the CR Yamabe

‡ow. There are geometric quantities (for example the Tanaka-Webster scalar curvature) which satisfy equation (2.20) under the CR Yamabe ‡ow in a closed CR 3-manifold. Indeed, these estimates can be used for understanding the singular models of positive Tanaka-Webster

curvature under the CR Yamabe ‡ow. In particular, this estimate should be useful in understanding the Yamabe solitons which one expects to model …nite time singularities of the CR Yamabe ‡ow.

Now we deal with c = 2 in (2.19). In particular, it follows that for u = R

(2.20) @R

@t = 4 _tR + 2R²:

Then we have the CR Li-Yau-Hamilton inequality of the Yamabe ‡ow (5.15). That is Theorem 2.12. ([CCK]) Let (M; J; ) be a closed spherical pseudohermitian 3-manifold with positive Tanaka-Webster curvature and vanishing torsion. Then under the CR Yamabe ‡ow (5.15),

(2.21) 4jr^bRj²

R²

R_t R

1 t 0:

Furthermore, we get a subgradient estimate of logarithm of the positive Tanaka-Webster curvature

jr^bRj² R²

1 4t for all t 2 (0; T ):

Remark 2.1. 1. Let (M; J; ) be a closed pseudohermitian 3-manifold. We call a CR structure J spherical if Cartan curvature tensor Q11 vanishes identically. Here

Q₁₁ = 1

6W₁₁+ i

2W A₁₁ A_11;0 2i

3A_11;^_₁₁:

Note that (M; J; ) is called a closed spherical pseudohermitian 3-manifold if J is a spherical structure. We observe that the spherical structure is CR invariant and a closed spherical pseudohermitian 3-manifold (M; J; ) is locally CR equivalent to (S³; bJ ; b):

2. If (M; J; ) is a closed pseudohermitian 3-manifold with A11 = 0, then R₀(x; 0) = 0 by the CR Bianchi identity. In additional if (M; J ) is spherical, then under the CR Yamabe

‡ow (5.15), R0(x; t) = 0 for all t.

By Chow connectivity theorem, there always exists a Legendrian curve joining any two points p and q, so the distance is …nite. Now integrating (2.21) over ( (t); t) of a Legendrian path : [t₁; t₂] ! M joining points x¹; x₂ in M; we obtain the following CR Harnack inequality for the positive Tanaka-Webster curvature under the CR Yamabe ‡ow.

Corollary 2.13. Let (M; J; ) be a closed spherical pseudohermitian 3-manifold with positive Tanaka-Webster curvature and vanishing torsion. Then under the CR Yamabe ‡ow (5.15), we have for all points x1, x2 in M and times t1 < t₂,

R(x₁; t₁) (t2

t₁)⁶⁴⁶³R(x₂; t₂) exp(1 2L);

where

L = inf Z t2

t1

(R + 1

8j _ j²J; _(t))dt

and the in…mum is taken over all Legendrian paths with (t1) = x₁ and (t2) = x₂.

Finally, in the papers of B. Chow and R. Hamilton [CH], L. Ni and L.-F. Tam [NT1]

proved the nonlinear trace Li-Yau-Hamilton inequality for the coupled the Ricci ‡ow and Kaehler Ricci ‡ow, respectively. Here we present its application of Theorem 2.7 to obtain the nonlinear version of Li-Yau-Hamilton inequality for CR Lichnerowicz-Laplacian heat equation (2.5) coupled with the CR Yamabe ‡ow (5.15).

Theorem 2.14. ([CCF]) Let (M; J;⁰) be a closed spherical pseudohermitian 3-manifold with positive Tanaka-Webster curvature and vanishing torsion. Let ₁₁(x; t) be a positive symmetric (1; 1)-tensor satisfying the CR Lichnerowicz-Laplacian heat equation (2.5) coupled with the CR Yamabe ‡ow (5.15) on M [0; T ) and _11;0 = 0 for all t. Then

Z_R := Z + RH 0 on M [0; T ) for k₁ = 4. In particular, taking V = 0

2 _{b 11}+ (R +1

t)H 0 and

(2.22) @

@t ¹¹+ 2(R + 1

t)H 0 with H = h¹¹ ₁₁:

As a consequence of Theorem 2.14 with ₁₁ = R₁₁ = Rh₁₁; we have the following trace Harnack inequality for the CR Yamabe ‡ow (5.15) which turns out to be a special case of the linear Harnack inequality for the CR Lichnerowicz-Laplacian heat equation (2.5) coupled with the CR Yamabe ‡ow (5.15) on a closed strictly pseudoconvex spherical CR 3-manifold.

This is the same as (2.21).

Corollary 2.15. Let (M; J;

0

) be a closed spherical pseudohermitian 3-manifold with positive Tanaka-Webster curvature and vanishing torsion. Then we have the following trace Harnack inequality for the CR Yamabe ‡ow (5.15)

@

@t(t²R) 0:

Finally, we point out that, by applying Hamilton’s general method, one can obtain the Harnack inequalities ([H1], [C]) to the CR Yamabe ‡ow.

Theorem 2.16. ([CCF]) Let (M; J;

0

) be a closed spherical pseudohermitian 3-manifold with positive Tanaka-Webster curvature and vanishing torsion. Then under the CR Yamabe ‡ow

(2.23) @R

@t +2R

t + 2hr^bR; ViJ; + 3

40RkV k²J; 0 for any V 2 T^1;0(M ) :

It is our hope that the similar nonlinear trace Li-Yau-Hamilton (2.22) holds as well for the torsion ‡ow (2.6) in a closed pseudohermitian 3-manifold.

3. Preliminary

First we introduce some basic materials in a pseudohermitian (2n + 1)-manifold (see [L1], [L2] for more details). Let (M; ) be a (2n+1)-dimensional, orientable, contact manifold with contact structure . A CR structure compatible with is an endomorphism J : ! such that J² = 1. We also assume that J satis…es the following integrability condition: If X and Y are in , then so are [J X; Y ] + [X; JY ] and J ([J X; Y ] + [X; JY ]) = [J X; J Y ] [X; Y ].

Let fT; Z ; Z g be a frame of T M C, where Z is any local frame of T^1;0; Z = Z 2 T^0;1 and T is the characteristic vector …eld. Then ; ; , which is the coframe dual to fT; Z ; Z g, satis…es

(3.1) d = ih ^

for some positive de…nite hermitian matrix of functions (h ), if we have this contact struc- ture, we also call such M a strictly pseudoconvex CR (2n + 1)-manifold.

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 T^1;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 )ⁿ, we get an inner product on the space of sections of each tensor bundle.

The pseudohermitian connection of (J; ) is the connection r on T M C (and extended to tensors) given in terms of a local frame Z 2 T^1;0 by

rZ = Z ; rZ = Z ; rT = 0;

where are the 1-forms uniquely determined by the following equations:

d = ^ + ^ ;

0 = ^ ;

0 = + ;

We can write (by Cartan lemma) = A with A = A . The curvature of Webster- Stanton connection, expressed in terms of the coframe f = ⁰; ; g, is

= = d! ! ^ ! ;

0 = ⁰ = ₀ = ⁰ = ₀⁰ = 0:

Webster showed that can be written

= R ^ + W ^ W ^ + i ^ i ^

where the coe¢ cients satisfy

R = R = R = R ; W = W :

Here R is the pseudohermitian curvature tensor, R = R is the pseudohermitian Ricci curvature tensor and A is the torsion tensor. Furthermore, we de…ne the bi-sectional curvature

R (X; Y ) = R X X Y Y and the bi-torsion tensor

T (X; Y ) := i(A X Y A X Y ) and the torsion tensor

T or(X; Y ) := h T (X; Y ) = i(A X Y A X Y ) for any X = X Z ; Y = Y Z in T1;0:

We will denote components of covariant derivatives with indices preceded by comma;

thus write A ; . The indices f0; ; g indicate derivatives with respect to fT; Z ; Z g. For derivatives of a scalar function, we will often omit the comma, for instance, u = Z u; u = Z Z u ! (Z )Z u:

For a smooth real-valued function u, the subgradient r^b is de…ned by r^bu 2 and hZ; r^buiL = du(Z) for all vector …elds Z tangent to contact plane. Locally r^bu = P u Z + u Z . We also denote u0 = Tu.

We can use the connection to de…ne the subhessian as the complex linear map (r^H)²u : T_1;0 T_0;1 ! T^1;0 T_0;1

by

(r^H)²u(Z) =r^Zr^bu:

In particular,

jr^buj² = 2u u ; jr²buj² = 2(u u + u u ):

Also

bu = T r (r^H)²u =P

(u + u ):

The Kohn-Rossi Laplacian b on functions is de…ned by

b' = 2@_b@_b' = ( _b+ inT )' = 2' and on (p; q)-forms is de…ned by

b = 2(@_b@_b+ @_b@_b):

Next we recall the following commutation relations ([L1]). Let ' be a scalar function and

= be a (1; 0) form, then we have

' = ' ;

' ' = ih '₀;

'₀ ' ₀ = A ' ;

;0 ; 0 = _; A A _; ;

;0 ; 0 = ; A + A _; ;

and

; ; = iA iA ;

; ; = ih A ih A ;

; ; = ih _;0+ R :

Moreover for multi-index I = ( 1; :::; _p) ; J = ₁; :::; _q ; we denote I( k = ) = ( ₁; :::; _{k 1}; ; _k+1; :::; _p) : Then

IJ ; IJ ; = i

Pp k=1

I( k= )JA _{k I( k= )J}A _k i

Pq

k=1 IJ( k= )h ^k A _IJ( k= )h ^k A ; and

IJ ; IJ ; = ih _{IJ ;0}+

Pp

k=1 I( k= )JR

k +

Pq

k=1 IJ( k= )R k

IJ ;0 IJ ; 0 = A _{IJ ;} Pp k=1

A _k ; I( k= )J + Pq k=1

A _;

k IJ( k= ):

4. CR Matrix Li-Yau-Hamilton Harnack Inequality

In the seminal paper [LY], P. Li and S.-T. Yau established the parabolic Li-Yau Harnack estimate for the positive solution u(x; t) of the time-independent heat equation

@u (x; t)

@t = u (x; t)

in a complete Riemannian l-manifold with nonnegative Ricci curvature. Here is the Laplace-Beltrami operator. Later in [H2], Richard Hamilton extended the Li-Yau estimate to the full matrix version of the Hessian estimate of u under the stronger assumptions that M is Ricci parallel and of nonnegative sectional curvature. Furthermore, Hamilton [H1] proved the matrix Harnack inequality for solutions to the Ricci ‡ow when the curvature operator is nonnegative. This inequality is called the “Li-Yau-Hamilton”type estimates. Since then, there are many additional works in this direction which cover various di¤erent geometric evolution equations such as the mean curvature ‡ow [H2], the Kähler-Ricci ‡ow [Ca], the Yamabe ‡ow [C], etc.

Along this line with method of Li-Yau gradient estimate, H.-D. Cao and S.-T. Yau ([CY]) studied the heat equation

(4.1) @u (x; t)

@t = Lu(x; t)

in a closed l-manifold with a positive measure and a subelliptic operator with respect to the sum of squares of vector …elds L = Ph

i=1X_i² Y; h l; with Y = Ph

i=1c_iX_i where X₁; X₂; :::; X_h are smooth vector …elds satisfying Hörmander’s condition : the vector …elds together with their commutators up to …nite order span the tangent space at every point of M: Suppose that [Xi; [X_j; X_k]] can be expressed as linear combinations of X 1; X₂; :::; X_h and their brackets [X1; X₂]; :::; [X_{l 1}; X_h]: They showed that the gradient estimate for the positive solution u(x; t) of (4.1) on M [0;1):

Recently in the paper of [CKL1], we obtained the CR Cao-Yau type gradient estimate for the positive solution u(x; t) of the CR heat equation

(4.2) @u (x; t)

@t = _bu (x; t)

in a closed pseudohermitian (2n + 1)-manifold (M; J; ) of nonnegative Tanaka-Webster curvature and vanishing torsion. Here b is the time-independent sub-Laplacian operator.

In this part, we will derive the following CR analogue of matrix Li-Yau-Hamilton inequality for any positive solution u to (4.2).

Theorem 4.1. Let M be a closed pseudohermitian (2n + 1)-manifold with nonnegative bi- sectional curvature and nonnegative bi-torsional tensor. Let u be the positive solution of the CR heat equation (4.2). In addition if the positive solution u satis…es the purely holomorphic Hessian operator P u = 0: Then

(4.3) (u + u ) + 1

2[(u V + u V ) + uV V ] t 12

ju⁰j²

u h + 4

tuh 0

for t > 0 and any vector …led V = V of type (1; 0) on M: Here P is the purely holomorphic Hessian operator (De…nition 4.1):

Corollary 4.2. The CR matrix Li-Yau-Hamilton inequality (4.3) holds in a closed pseudo- hermitian (2n + 1)-manifold of nonnegative bisectional curvature and vanishing torsion.

Remark 4.1. If we choose the optimal V = ru=u and take the trace of (4.3), we recapture the following CR Li-Yau gradient estimate which was derived by Chang-Kuo-Lai in [CKL1]

and [CKL2] :

(4.4) @

@tu 1 4

kruk² u

nt 12

ju⁰j² u +4n

t u 0:

When the manifold is complete noncompact, we will need to use the CR Li-Yau Harnack inequality (4.29) and Li-Tam mean value inequality (4.32) in the proof of the CR matrix Li-Yau-Hamilton inequality (4.3). However, the proof of both inequalities rely on CR Li-Yau gradient estimate (4.5). We refer to [CN] for some details.

As shown in section 4:2; the proof of CR Li-Yau gradient estimate (4.5) relies on the CR sub-Laplacian comparison property (4.27) and the extra u0-growth property (see appendix in [CFTW]) with ju⁰j ^C_tu that has no analogue in the Riemannian case. In particular, both properties holds in a standard Heisenberg group Hⁿ which is ‡at and vanishing torsion.

However, both properties are wild open in a general complete noncompact pseudohermitian (2n + 1)-manifold.

Then we are able to derive the following CR Li-Yau gradient estimate on Hⁿ:

Theorem 4.3. Let (Hⁿ; J; ) be the standard (2n + 1)-dimensional Heisenberg group. If u(x; t) is the positive solution of the CR heat equation (4.2) on Hⁿ [0;1). Let ' = ln u;

for any < 1; then there exists a positive constant C depending on such that (4.5) jr^b'j²+ '_t+ t'²₀ ^C_t:

By applying Theorem 4.3, we have the following CR Liouville-type theorem for a positive pseudoharmonic function u on (Hⁿ; J; ) which recaptured the Liouville theorem due to Chang-Kuo-Tie ([CKT]) and Koranyi and Stanton ([KS]) by a di¤erent method.

Corollary 4.4. Let (Hⁿ; J; ) be the standard (2n + 1)-dimensional Heisenberg group. If u(x; t) is the positive smooth function with _bu = 0; then u(x; t) is constant. That is, there does not exist any positive nonconstant pseudoharmonic function in Hⁿ.

From the previous discuss and Theorem 4.1, we have the CR matrix Li-Yau-Hamilton inequality in (Hⁿ; J; ) as in section 4:2:

Theorem 4.5. Let (Hⁿ; J; ) be the standard (2n + 1)-dimensional Heisenberg group. If u(x; t) is the positive solution of the CR heat equation (4.2) on Hⁿ [0;1). Then the CR matrix Li-Yau-Hamilton inequality (4.3) holds.

Remark 4.2. We observe that from the proof of Theorem 4.5 that the CR matrix Li-Yau- Hamilton inequality (4.3) still holds in a complete noncompact pseudohermitian manifold whenever both the CR sub-Laplacian comparison property (4.27) and the u0-growth property hold. We should point out that the extra u0-growth property is equivalent to (4.35) that has no analogue in Kähler manifolds.

By applying Theorem 4.5 to the heat kernel H(x; y; t) with V = ^rH_H and observe that the well-known asymptotic of H(x; o; t) ( [V], [Ga], [B], [Le], [T], [BBN], etc)

t log H(x; o; t)! 1 4r²(x)

as t ! 0: Here r(x) be the Carnot-Carathéodory distance function to the origin o 2 Hⁿ. We have the following complex Hessian comparison theorem for r on Hⁿ. Such a Hessian comparison property seems to be new in the standard (2n + 1)-dimensional Heisenberg group Hⁿ:

Corollary 4.6. Let (Hⁿ; J; ) be the standard (2n + 1)-dimensional Heisenberg group. Then in the sense of distribution, we have

[(r²(x)) + (r²(x)) ] (16 + C₀)h (x)

for some constant C0. In particular, we recapture the sub-Laplacian comparison property

br²(x) (16 + C₀)n in the Heisenberg group.

In the following, in section 4:1, we prove the CR matrix Li-Yau-Hamilton inequality for the CR heat equation via methods developed as in [LY], [CKL1] and [CN]. In section 4:2, we prove a CR Li-Yau gradient estimate in the standard (2n + 1)-dimensional Heisenberg group.

Combining this with Theorem 4.1, we have the CR matrix Li-Yau-Hamilton inequality and Hessian comparison property in the standard (2n + 1)-dimensional Heisenberg group Hⁿ.

4.1. CR Matrix Li-Yau-Hamilton Inequality. Let u(x; t) be the positive solution of the CR heat equation (4.2). For the CR Li-Yau gradient estimate as in the paper [CKL1], we observe that one of di¢ culties is to deal with CR Bochner formula ( 4.16) which involving a term hJr^b'; r^b'₀i that has no analogue in the Riemannian case. In order to overcome this di¢ culty, we introduce a new scalar Harnack quantity G = t[jr^b'j² + '_t+ t'²₀] with ' = ln uby adding an extra term t'²₀to jr^b'j²+ '_twhich was appeared in Li-Yau estimate ([LY]). See section 4:2 for more details.

Now we want to …nd the right quantity for the CR matrix Li-Yau-Hamilton inequality. By comparing the Harnack quantity in [CN] in case of Kähler manifolds, we de…ne the matrix Harnack quantity

(4.6) N = 1

2(u + u ) + 2u

th bu u

u atju⁰j² u h

by adding an extra term F := at^ju_u^0j2h in which positive constants a and b to be deter- mined later (say a = ₂₄¹and b = ¹₄).

De…nition 4.1. (i) ([GL]) De…ne the purely holomorphic Hessian operator P : P ' := 2i(A ' )

and the purely holomorphic Poisson operator Q :

Q' := h (P ') = 2i(A ' )

for any smooth function ': Note that P ' = 0 = Q' for any smooth function ' if A = 0 on M:

Lemma 4.1. Let u(x; t) be the positive solution of the CR heat equation (4.2). Then

1

2(u + u ) satis…es the following : 1

2

@

@t ^b u + u = 2R u R u R u + C ;

where

C := i A _; u A _; u h + i A u A u h

+in A u A u + in A _; u A _; u

:= (Re Qu)h + n(Re P u):

Note that trC = h C = 0: In particular we have C₁₁= 0 for n = 1: In addition if the positive solution u satis…es P u = 0 which is the case when the torsion is vanishing, then u satis…es the following CR Lichnerowicz-Laplacian heat equation ([CCF]) :

@

@t ^b u = 2R u R u R u :

Proof. Note that

@

@t b u + u

= _@t^@ u + u _b u + u

= [( _bu) _bu ] + [( _bu) _bu ]:

(i) We …rst compute [( bu) _bu ] :By de…nition, we have

(4.7) ( _bu) = (u + u ) = u + u :

Compute

(4.8)

u = (u ih u ₀ R u )

= u ih u ₀ R _; u R u

= u ih u ₀ R _; u R u

= u + i u h A u h A

+i (nu A u h A )

ih u ₀ R _; u R u

= u + ih u ₀+ R u

+i u h A u h A

+i (nu A u h A )

ih u ₀ R _; u R u

= u + ih u ₀ ih u ₀

R _; u R u + R _; u + R u

+i u h A u h A + i (nu A u h A ) :

Here we have use commutation relations

u = u + i u h A u h A

+i (nu A u h A )

and

u = u + ih u ₀+ R u

= u + ih u ₀ + R _; u + R u : Similiar, we have

(4.9)

u = u + ih A _; u + ih A _; u

ih A _; u ih A u i (nA u ) + i (h A u )

+R u + R u

ih u₀ + ih u ₀ It follow from (4.7), (4.8) and (4.9) that

(4.10)

( _bu) _bu = +2R u R u R u

+ R _; R _; u

+ih u ₀ ih u ₀ ih u₀ + ih u ₀

+i u h A u h A + i (nu A u h A )

+ih A _; u + ih A _; u ih A _; u ih A u i (nA u ) + i (h A u )

By CR Bianchi identity ([L1]) and commutation relation, the third line of RHS in (4.10) becomes

R _; R _;

= R _; R _;

= R _; R _;

= iA ; h iA ; h + iA _; h + iA _; h

= iA _; h iA _; h + iA _; h + inA _; and the fourth line becomes

ih u ₀ ih u ₀ ih u₀ + ih u ₀

= iu ₀ iu ₀ iu₀ + iu ₀

= iu ₀ iu₀

= iA _; u iA u iA u iA _; u

(ii) We compute [( bu) _bu ] by take the conjugate of [( bu) _bu ] and then switch index and :

Now we arrange all the torsion terms together in (i) and (ii), then we are done.

Note that it follows from commutation relation ([CKL1]) that

bu₀ = ( _bu)₀+ 2h

(A u ) + (A u ) i : Hence

[ b; T] u = 2 Im Qu:

Proof of Theorem 4.1 :

Proof. As in [H2], it su¢ ces to prove that the Hermitian symmetric (1; 1)-tensor

N = 1

2(u + u ) + 2u

th bu u

u F h 0

for t > 0 and some constant a and b to be determined. Here F := atju⁰j²

u : Now we …rst compute

@

@t b

u u

u = _@t^{@ u u}_{u b} ^{u u}_u

= _u2¹ bu u u +_u¹ ( _bu) u +_u¹u ( _bu) _{b u}¹u u and

b 1

uu u = _u2^bu + _u⁴3u u u u + ¹_{u b} u u

2

u²u u u _u²2u u u :

Hence

@

@t b

u u

u = ²_uu u _u²u u _u²3 jruj²u u

+_u²2u u u + _u²2u u u

+_u¹(( _bu) _b(u )) u + ¹_u ( _bu) _b u u :

Therefore by using Lemma 4.1, we have

@

@t b N

= 2R u R u R u + C

+b _u²u u + ²_uu u +_u²₃ jruj²u u _u²₂u u u + u u b_u²2u u u + u u 2_t^u2h

b_u¹(( _bu) _b(u )) u b_u¹ ( _bu) _b u u

@

@t b F h :

Observe that (4.11)

1

u(( _bu) _b(u )) u

= _u¹ (u + u u u ) u

= _u¹ (u ih u 0 R u + u ih u0 inA u + ih A u u u ) u

= _u¹ iu ₀u R u u iu₀ u inA u u + iA u u

= _u¹R u u 2i^u0_u^u (n 2) i¹_uA u u : Thus

@

@t b N

= 2R N R N R N + C

+2bR ^{u u}_u + b (n 2) i¹_uA u u b (n 2) i_u¹A u u +^2b_u u ^{u u}_u u ^{u u}_u

+2b ¹_uu u _u¹2u u u _u¹2u u u +2bi^u_u⁰2u u 2bi^u_u⁰u _@t^@ _b F h +bjruj^{2 u u}_u^{3 2u}_t²h

+2bi^u0_u^u 2bi^u0_u^u : Note that we can rewrite N as following :

N = u 1

2iu₀h + 2u

th bu u

u F h :

Then we replace u = N +^iu0₂^h 2^u_th + b^{u u}_u + F h into third and forth line of RHS as above, we have

2b _u¹u u _u¹₂u u u _u¹₂u u u +ib^2u_u2⁰u u ib^2u_u⁰u _@t^@ _b F h

= ^2b_uN N ^8b_tN + 8b_t^u2h + (b³ 2b²)jruj^{2 u u}u³

+ (8b 8b²)¹_t^{u u}_u + b^u_2u²⁰h

+b^{2 2u u}_u2 N + b^{2 2u u}_u2 N b_u²2u u N b_u²2u u N +^4b_uF N + ^2b_uF²h + 4 (b² b)^{F u u}_u2

2b⁴_tF h _@t^@ b F h :

Finally one obtains

(4.12)

@

@t b N

= 2R N R N R N + C

+2bR ^{u u}_u + b (n 2) i¹_uA u u b (n 2) i_u¹A u u +^2b_u u ^{u u}_u u ^{u u}_u + ^2b_uN N ^8b_tN

+b^{2 2u u}_u2 N + b^{2 2u u}_u2 N _u^2b2u u N _u^2b2u u N + ^4b_uF N

@

@t b F h + ^2b_uF²h + 4 (b² b)^{F u u}_u2

+ (8b 2)_t^u2h + b^u_2u²⁰h

+ (8b 8b²)¹_t^{u u}_u + (b³ 2b²+ b)jruj^{2 u u}_u³

8b

tF h + 2bi^u0_u^u 2bi^u0_u^u

Note the …rst and second line of RHS are positive by curvature assumption. The third and fourth line are nonnegative while we apply on null vector of N .

In the following we determined F to make the rest terms nonnegative. First observe that

@

@t b

u²₀

u = ^2u_u⁰[T; ]u ^2kru_u^0k2 + ^4u0hru_u2^0;rui 2u²₀^kruk_u3 ²

= 2 ^ru0

u¹²

u0ru u³²

2

;

where we use the fact that [T; ]u = 2 Im Qu = 0which is always true if P u = 0. The last four lines of (4.12) become

(4.13)

b

2 a (1 + 8b) ^u_u²⁰h + 2at ^ru0

u¹²

u0ru u³²

2

h +2 a²t²b^u_u⁴⁰3h + 2(^{b2 b})

pb atp

b^{u20u u}_u3 + (^{b2 b})²

b

jruj² 2

u u u³

+ (8b 2)_t^u2h

+2bi^u0_u^u 2bi^u0_u^u + 8b (1 b)^{u u}_tu :

Note that the second line above is a complete square. To handle the last term, we have following

(4.14)

2bi^u0_u^u 2bi^u0_u^u + 8b (1 b)^{u u}_tu

= 2bi^u0 p^u0uu^u

puu 2bi^u0

u0u

puu

puu + 8b (1 b) ^{u u}_tu

= b "^u0 p^u0uu^u 2

"ip^uu "^u0

u0u

puu +²_"ip^uu

b"^{2 u0} p^u0uu u

u₀ ^u0u_u pu

4b

"² u u

u + 8b (1 b)^{u u}_tu : By taking "² = ^4at_b ; we have

2at ^ru0

u¹²

u0ru u³²

2

h b"^{2 u0} p^u0uu^u

u₀ ^u0u_u pu

= 2at ^ru0

u¹²

u0ru u³²

2

h 4at^u0 p^u0uu u

u₀ ^u0u_u pu

0:

Then by applying (4.13) and (4.14)

b

2 a (1 + 8b) ^u_u²⁰h + 2at ^ru0

u¹²

u0ru u³²

2

h +2 a²t²b^u_u⁴⁰3h + 2(^{b2 b})

pb atp b^u

2 0u u

u³ + (^{b2 b})²

b

jruj² 2

u u u³

+ (8b 2)_t^u2h

+2bi^u0_u^u 2bi^u0_u^u + 8b (1 b)^{u u}_tu

b

2 a (1 + 8b) ^u_u²⁰h + (8b 2)_t^u2h + 8b (1 b) ^b_a^{2 u u}_tu

= 0

when we choose a; b such that

b

2 a (1 + 8b) = 0;

8b 2 = 0;

8b (1 b) ^b_a² = 0:

That is

a = 1

24 and b = 1 4: Hence from (4.12)

(4.15)

@

@t b N

2R N R N R N +_2u¹ R u u

+_2u¹ u ^{u u}_u u ^{u u}_u + _2u¹N N ²_tN

+¹₈^{u u}_u2 N + ¹₈^{u u}_u2 N _2u¹2u u N _2u¹2u u N + ¹_uF N +C + _4u¹ (n 2) i[A u u A u u ]:

which is nonnegative while we apply on null vector of N we assume nonnegative bisectional curvature, nonnegative bi-torsion tensor and nonnegative C as well.

4.2. The CR Gradient Estimate and Harnack inequality in Heisenberg Groups.

In this section, by using the method of CR Li-Yau gradient estimate ([LY], [CKL1]) and CR Bochner formula (4.16), we derive a CR gradient estimate and CR Harnack inequality for the positive solution of the CR heat equation (4.2) in (2n + 1)-dimensional Heisenberg group.

We …rst recall the following CR version of Bochner formula in a complete pseudohermitian (2n + 1)-manifold.

Lemma 4.2. ([Gr]) For a smooth real-valued function ';

(4.16)

1

2 bjr^b'j² = j(r^H)²'j²+hr^b';r^{b b}'i + 2 hJr^b';r^b'₀i +[2Ric (n 2)T or] ((r^b')_C; (r^b')_C) : Here (r^b')_C = ' Z is the corresponding complex (1; 0)-vector of r^b'.

Since

j(r^H)²'j² = 2P

; (j' j²+j' j²) 2P

j' j² _2n¹ ( _b')²+ ⁿ₂'²₀