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CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions

Shu-Cheng Chang Department of Mathematics and TIMS, NTU (joint work with Ting-Jung Kuo and Jingzhu Tie)

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 1 / 55

Overview

Introduction

1

Gradient estimate in Riemannian case

2

Main results

1

Sub-gradient estimate for positive pseudoharmonic functions

2

CR analogue of Liouville-type theorem for positive pseudoharmonic functions

The CR Bochner-Type Estimate The Proofs The CR sub-Laplacian comparison property

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 2 / 55

Gradient estimate in Riemannian manifold

S.-Y. Cheng and S.-T. Yau derived a well known gradient estimate for positive harmonic functions in a complete noncompact Riemannian manifold.

Theorem 2.1

Let M be a complete noncompact Riemannian m-manifold with Ricci curvature bounded from below by K (K 0). If u (x) is a positive harmonic function on M, then there exists a positive constant C = C(m) such that

jruj

u

2

C( p

K + 1 R ) (2.1)

• n the ball B (R) .

As a consequence, the Liouville theorem holds for complete noncompact Riemannian m-manifolds with Ric 0.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 3 / 55

CR Yau’s gradient estimates

Yau’s Geometric Analysis :

1

Bochner formula

2

Laplacian comparison

3

maximum principle

CR Geometric and Analytic aspects :

1

Riemannian Ricci curvature $pseudohermitian Ricci curvature tensor (Rij) and pseudohermitian torsion (A11)

2

Problem 2.2

Sub-Laplacian ∆b is degenerated along the missing dirction T by comparing the Riemannian Laplacian ∆.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 4 / 55

CR Yau’s gradient estimates

By modifying the arguments of Yau, we are able to derive CR version of Yau’s gradient estimate on (M, J, θ) with the CR sub-Laplacian comparison property. The same argument can be used to prove CR Li-Yau gradient estimate for positive solutions of the CR heat equation (Chang-Tie-Wu).

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 5 / 55

CR Yau’s gradient estimates

De…nition 2.3

Let (M, J, θ) be a complete noncompact pseudohermitian

(2n + 1)-manifold with (2Ric (n 2)Tor) (Z, Z) 2k jZj2

(2.2) for all Z 2 T1,0, and k is an nonnegative constant. We say that

(M, J, θ) satis…es the CR sub-Laplacian comparison property if

there exists a positive constant C0 = C0(k, n) such that ∆br C0(1 r +

p

k) (2.3) in the sense of distributions. Here r (x) is the Carnot-Carathéodory distance from a …xed point x0 2 M.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 6 / 55

CR Yau’s gradient estimates

It is still not clear that whether the CR sub-Laplacian comparison property holds in a complete noncompact pseudohermitian (2n + 1)-manifold (M, J, θ). However, it can be shown that the CR sub-Laplace comparison property holds in the standard Heisenberg (2n + 1)-manifold

(Hn, J, θ).

Proposition 2.4

Let (Hn, J, θ) be a standard Heisenberg (2n + 1)-manifold. Then there exists a constant CHn

1

> 0

∆brHn CHn

1

rHn . (2.4)

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 7 / 55

Weak sub-gradient estimate

Theorem 3.1

Let (M, J, θ) be a complete noncompact pseudohermitian

(2n + 1)-manifold with (2Ric (n 2)Tor) (Z, Z) 2k jZj2

max

• Aαβ
• ,
• Aαβ,¯

α

• k1

for all Z 2 T1,0 and k 0, k1 > 0. Assume that (M, J, θ) satis…es the CR sub-Laplacian comparison property. If u (x) is a positive pseudoharmonic function on M (i.e. ∆bu = 0).

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 8 / 55

Weak sub-gradient estimate

Then there exists a small constant ˜ b = ˜ b(n, k, k1) > 0 and C4 = C4(k, k1, k2) such that for any 0 < b ˜ b,

jrbuj2

u2

+ bu2

u2 < (n + 5)2 5

• k + n (1 + b) k1 + 2

b + C4 R

• (3.1)
• n the ball B (R) of a large enough radius R which depends only on
• b. Here u0 = Tu, and T is the Reeb vector …eld on M.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 9 / 55

Sub-gradient estimate

In order to derive Liouville-type theorem for positive pseudoharmonic functions (i.e. ∆bu = 0), we need to show a stronger sub-gradient estimate To show the stronger sub-gradient estimate, we have to put the condition

[∆b, T]u = 0.

where T is the Reeb vector …eld.

De…nition 3.2

([GL]) Let (M, J, θ) be a pseudohermitian (2n + 1)-manifold. We de…ne the purely holomorphic second-order operator Q by Qu = 2i

n

∑

α,β=1

(A¯

α ¯ βuβ),α .

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 10 / 55

Sub-gradient estimate

Theorem 3.3

Let (M, J, θ) be a complete noncompact pseudohermitian

(2n + 1)-manifold with (2Ric (n 2)Tor) (Z, Z) 2k jZj2

for all Z 2 T1,0, and k 0. Assume that (M, J, θ) satis…es the CR sub-Laplacian comparison property. If u (x) is a positive pseudoharmonic function with

[∆b, T]u = 0

(3.2)

• n M.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 11 / 55

Sub-gradient estimate

Then for each constant b > 0, there exists a positive constant C2 = C2(k) such that

jrbuj2

u2

+ bu2

u2 < (n + 5 + 2bk)2

(5 + 2bk)

• k + 2

b + C2 R

• (3.3)
• n the ball B (R) of a large enough radius R which depends only on

b, k.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 12 / 55

CR Liouville theorem

As a consequence, let R ! ∞ and then b ! ∞ with k = 0 in (3.3),we have the following CR Liouville-type theorem.

Corollary 3.4

Let (M, J, θ) be a complete noncompact pseudohermitian

(2n + 1)-manifold with (2Ric (n 2)Tor) (Z, Z) 0

for all Z 2 T1,0. Assume that (M, J, θ) satis…es the CR sub-Laplacian comparison property. If u (x) is a positive pseudoharmonic function with

[∆b, T]u = 0.

Then u (x) is constant.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 13 / 55

Liouville-type theorem for Heisenberg manifold

Fact 3.5

It is shown that

[∆b, T] u = 4 Im[i

n

∑

α,β=1

(A¯

α ¯ βuβ),α ].

(3.4) If (M, J, θ) is a complete noncompact pseudohermitian

(2n + 1)-manifold with vanishing torsion. Then [∆b, T] u = 0.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 14 / 55

CR Liouville-type theorem

Fact 3.6

On (Hn, J, θ), we have Rα ¯

β = 0, and Aαβ = 0.

Moreover,

[∆b, T] = 0

holds on (Hn, J, θ).

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 15 / 55

CR Liouville-type theorem

By applying Corollary 3.4 and Proposition 2.4, we have the following CR Liouville-type theorem for a positive pseudoharmonic function u on (Hn, J, θ)

Corollary 3.7

There does not exist any positive nonconstant pseudoharmonic function in a standard Heisenberg (2n + 1)-manifold (Hn, J, θ). Koranyi and Stanton proved the Liouville theorem in (Hn, J, θ) by a di¤erent method via heat kernel.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 16 / 55

The CR Bochner-Type Estimate

Firstly, we recall the Bochner formula in Riemannian geometry.

Fact 4.1

For a real function ϕ, we have ∆ jrϕj2

=

2

• r2ϕ
• 2 + 2 hrϕ, r∆ϕi + 2Ric (rϕ, rϕ) .

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 17 / 55

CR Bochner Formula

Lemma 4.2

For a real function ϕ, ∆b jrbϕj2

=

2

• rH2 ϕ
• 2

+ 2 hrbϕ, rb∆bϕi + (4Ric 2 (n 2) Tor) ((rbϕ)C , (rbϕ)C ) + 4 hJrbϕ, rbϕ0i .

(4.1) where (rbϕ)C = ϕ¯

αZα is the corresponding complex (1, 0)-vector

• f rbϕ.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 18 / 55

Let f = ln u, where ∆u = 0. In Yau’s method, one can estimate ∆(η (x) jrf (x)j2) on B (2R) via Bochner formula and Laplacian comparison. At the end, by applying the maximum principle to η (x) jrf (x)j2, one obtains gradient estimate. Similarly, one has to apply the CR Bochner formula to f , where f = ln u and ∆bu = 0. However, one di¢culty is to deal with a term hJrbf , rbf0i appearing in CR Bochner formula that has no analogue to the Riemannian case.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 19 / 55

CR Bochner Inequality

Lemma 4.3

For a real function ϕ and any ν > 0, we have ∆b jrbϕj2

• 4

n

∑

α,β=1

• ϕaβ
• 2 +

n

∑

α,β=1,α6=β

• ϕa ¯

β

• 2

!

+ 1

n (∆bϕ)2

+

nϕ2

0 + 2 hrbϕ, rb∆bϕi 2ν jrbϕ0j2

+

• 4Ric 2 (n 2) Tor 4

ν

((rbϕ)C , (rbϕ)C ) , where (rbϕ)C = ϕ¯

αZα is the corresponding complex (1, 0)-vector

• f rbϕ.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 20 / 55

CR Bochner Inequality

Proof.

We …rst have

j(rH)2ϕj2 =

2(∑n

α,β=1 jϕαβj2 + ∑n α,β=1 α6=β

jϕαβj2) + 1

2 ∑n α=1 jϕαα + ϕααj2 + n 2 ϕ2

• 2(∑n

α,β=1 jϕαβj2 + ∑n α,β=1 α6=β

jϕαβj2) + 1

2n (∆bϕ)2 + n 2 ϕ2 0.

On the other hand, for all ν > 0 4 hJrbϕ, rbϕ0i

• 4 jrbϕj jrbϕ0j
• 2

ν jrbϕj2 2ν jrbϕ0j2 .

Then the result follows easily from Lemma 4.2.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 21 / 55

The Test Function

In order to overcome this di¢culty

• 2ν jrbf0j2

, we introduce a real-valued function: F (x, t, R, b) : M [0, 1] (0, ∞) (0, ∞) ! R by adding an extra term f 2

0 (x) to jrbf (x)j2 as following

F (x, t, R, b) = t

• jrbf (x)j2 + btη (x) f 2

0 (x)

• n the Carnot-Carathéodory ball B (2R) with a constant b to be

determined. One applies ∆b to the function F, then ∆bf 2

0 = 2f0∆bf0 + 2 jrbf0j2

contributes a positive term jrbf0j2 which can be used to balance the term

• 2ν jrbf0j2

.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 22 / 55

Purely holomorphic second-order operator Q

Lemma 4.4

Let ϕ (x) be a smooth function de…ned on M. Then ∆bϕ0 (∆bϕ)0 = 2

n

∑

α,β=1

h Aαβϕ ¯

β

• ¯

α +

• A¯

α ¯ βϕβ

• α

i . That is 2 ImQϕ = [∆b, T] ϕ.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 23 / 55

Purely holomorphic second-order operator Q

We de…ne V (ϕ) =

n

∑

α,β=1

h Aαβϕ ¯

β

• ¯

α +

• A¯

α ¯ βϕβ

• α + Aαβϕ ¯

βϕ¯ α + A¯ α ¯ βϕβϕα

i .

Lemma 4.5

Let (M, J, θ) be a pseudohermitian (2n + 1)-manifold and u be a positive function with f = ln u. Suppose that 2 ImQu = [∆b, T] u = 0. Then V (f ) = 0. (4.2)

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 24 / 55

The Proofs

We recall a real-valued function F (x, t, R, b) : M [0, 1] (0, ∞) (0, ∞) ! R de…ned by F (x, t, R, b) = t

• jrbf j2 (x) + btη (x) f 2

0 (x)

• (5.1)

= t jrbf j2 + bt2η (x) f 2

ηF = tη jrbf j2 + bt2η2f 2

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 25 / 55

The Proofs

where η (x) : M ! [0, 1] is a smooth cut-o¤ function de…ned by η (x) = η (r (x)) = 1, x 2 B (R) 0, x 2 MnB (2R) such that

C

R η

1 2 η 0 0

(5.2) and

• η

00

• C

R2 , (5.3) where we denote ∂

∂r η by η

• 0. Here r(x) is the Carnot-Carathéodory

distance to a …xed point x0.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 26 / 55

The Proofs

Noting that from CR comparision property, one can have the following estimate on the cut o¤ function ∆bη

=

η

00 + η 0∆br

• C

R2 C R

• C1

R + C2

• C

R .

(5.4)

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 27 / 55

The Proofs

We …rst compute ∆b (ηF).

Proposition 5.1

Let (M, J, θ) be a complete noncompact pseudohermitian

(2n + 1)-manifold with CR sub-Laplacian comparison property.

Suppose that

(2Ric (n 2) Tor) (Z, Z) 2k jZj2

for all Z 2 T1,0, where k is an nonnegative constant.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 28 / 55

The Proofs

Then for all a 6= 0 tη∆b (ηF)

• 1

na2 (ηF)2 C R (ηF) + 2tη hrbη, rbFi 2tη2 hrbf , rbFi

+4t2η2

n

∑

α,β=1

• faβ
• 2 +

n

∑

α,β=1,α6=β

• fa ¯

β

• 2

!

+

• n bC

R

2b

na2 + bC R

(ηF)

• t2η2f 2

+

h

2(1+a)

na2

(ηF) 2k 4

b

i tη jrbf j2 + 4bt3η3f0V (f ) . (5.5)

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 29 / 55

The Proofs

Proposition 5.2

Let (M, J, θ) be a complete noncompact pseudohermitian

(2n + 1)-manifold with CR sub-Laplacian comparison property.

Suppose that

(2Ric (n 2) Tor) (Z, Z) 2k jZj2

for all Z 2 T1,0, where k is an nonnegative constant. Let b, R be …xed, and p (t) 2 B (2R) be the maximal point of ηF for each t 2 (0, 1].

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 30 / 55

The Proofs

Then at (p (t) , t) we have

• 1

na2 C R

(ηF)2 3C

R (ηF)

+4t2η2

n

∑

α,β=1

• faβ
• 2 +

n

∑

α,β=1,α6=β

• fa ¯

β

• 2

!

+

• n bC

R

2b

na2 + bC R

(ηF)

• t2η2f 2

+

h

2(1+a)

na2

(ηF) 2k 4

b C R

i tη jrbf j2

+4bt3η3f0V (f ) .

(5.6)

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 31 / 55

The Proofs

Proof.

Apply maximum principle to the maximum point p (t) for each t 2 (0, 1]. Since

(ηF) (p (t) , t, R, b) =

max

x2B(2R) (ηF) (x, t, R, b), at a critical

point (p (t) , t) of (ηF) (x, t, R, b), we have

rb (ηF) (p (t) , t, R, b) = 0. This implies that

Frbη + ηrbF = 0 (5.7) at (p (t) , t) .On the other hand, ∆b (ηF) (p (t) , t, R, b) 0 (5.8) at (p (t) , t) . Substituting (5.7) and (5.8) into (5.5), we obtain (5.6).

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 32 / 55

The Proofs

Next we prove Theorem 3.3. Since

[∆b, T] u = 0

, we have V (f ) = 0 (5.9)

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 33 / 55

The Proofs

Proof of Theorem 3.3 :

Proof.

We begin by substituting (5.9) into (5.6) in Proposition 5.2 at the maximum point (p(t), t). Hence

• 1

na2 C R

[(ηF)]2 3C

R [(ηF)]

+

• n bC

R

2b

na2 + bC R

(ηF)

• t2η2f 2

+

h

2(1+a)

na2

(ηF) 2k 4

b C R

i tη jrbf j2

+4t2

0η2 n

∑

α,β=1

• faβ
• 2 +

n

∑

α,β=1,α6=β

• fa ¯

β

• 2

! . (5.10)

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 34 / 55

The Proofs

Proof.

We claim at t = 1

(ηF) (p (1) , 1, R, b) <

na2

2 (1 + a)

• 2k + 4

b + C R

• (5.11)

for a large enough R which to be determined later. Here (1 + a) < 0 for some a to be chosen later (say 1 + a = 5+2bk

n

). We prove it by

• contradiction. Suppose not, that is

(ηF) (p (1) , 1, R, b)

na2

2 (1 + a)

• 2k + 4

b + C R

• .

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 35 / 55

The Proofs

Proof.

Since (ηF) (p (t) , t, R, b) is continuous in the variable t and

(ηF) (p (0) , 0, R, b) = 0, by Intermediate-value theorem there

exists a t0 2 (0, 1] such that

(ηF) (p (t0) , t0, R, b) =

na2

2 (1 + a)

• 2k + 4

b + C R

• .

(5.12) Now again we apply (5.10) at the point (p (t0) , t0), denoted by

(p0, t0). We have by using (5.12)

• 1

na2 C R

[(ηF) (p0, t0)]2 3C

R [(ηF) (p0, t0)]

+

• n bC

R

2b

na2 + bC R

(ηF) (p0, t0)

• t2η2f 2

+4t2

0η2 n

∑

α,β=1

• faβ
• 2 +

n

∑

α,β=1,α6=β

• fa ¯

β

• 2

! . (5.13)

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 36 / 55

The Proofs

Proof.

we compute 1

na2 C R

(ηF) (p0, t0) 3C

R

• =

h 1

na2 C R na2 2(1+a)

2k + 4

b + C R

3C

R

i

=

n

1 2(1+a)

• 2k + 4

b

C

R

h

na2 2(1+a)

• 2k + 4

b + C R

+

1 2(1+a) + 3

io (5.14) and

• n bC

R

2b

na2 + bC R

(ηF) (p0, t0)

• = n bC

R

2b

na2 + bC R na2 2(1+a)

2k + 4

b + C R

• =
• n +

4 1+a + 2bk 1+a

+ C

R

h

ab

1+a + na2b 2(1+a)

• 2k + 4

b + C R

i . (5.15)

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 37 / 55

The Proofs

Proof.

Now we choose a such that

(1 + a) < 4 + 2bk

n and then

• n +

4 1 + a + 2bk 1 + a

• > 0.

In particular, we let 1 + a = 5 + 2bk n . (5.16)

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 38 / 55

The Proofs

Proof.

Then for R = R(b, k) large enough, one obtains 1 na2 C R

• (ηF) (p0, t0) 3C

R

• > 0

and

• n bC

R 2b na2 + bC R

• (ηF) (p0, t0)
• > 0.

This leads to a contradiction with (5.13). Hence from (5.11) and (5.16)

(ηF) (1, p (1) , R, b) < (n + 5 + 2bk)2

2 (5 + 2bk)

• 2k + 4

b + C R

• .

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 39 / 55

Proof.

This implies max

x2B(2R)

• jrbf j2 + bηf 2
• (x) < (n + 5 + 2bk)2

2 (5 + 2bk)

• 2k + 4

b + C R

• .

When we …x on the set x 2 B (R), we obtain

jrbf j2 + bf 2

0 < (n + 5 + 2bk)2

2 (5 + 2bk)

• 2k + 4

b + C R

• n B (R). This completes the proof.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 40 / 55

The CR sub-Laplacian comparison property

Here, we sketch the proof of Prosition 2.4. For simplicity, for n = 1. In the setting, we write Z1 = 1

2(e1 ie2)

for real vectors e1, e2. It follows e2 = Je1.

Lemma 5.1

Let (M, J, θ) be a complete pseudohermitian 3-manifold. Then ∂r(∆br) + 2(∆br)2 2r0e1 + 2r2

0 + (W ImA11) = 0

(5.17) and r00e2 + r2

0e1 (Re A11)2 (Re A11)0 = 0.

(5.18) W is the pseudohermitian scalar curvature.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 41 / 55

The CR sub-Laplacian comparison property

In the Heisenberg manifold, we have the estimate

jr0e1j C

r2 From Lemma 5.1 and put W = 0, A11 = 0 and together with the estimate on r0e1, we get the following ode inequality ∂r(∆br) + 2(∆br)2 C r2 0. Then by ODE inequality, there exists a constant CH1

2

> 0 such that

∆br CH1

2

r .

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 42 / 55

Pesudohermitian (2n+1) manifold

Let (M, ξ) be a (2n + 1)-dimensional, orientable, contact manifold with contact structure ξ, dimR ξ = 2n. A CR structure compatible with ξ is an endomorphism J : ξ ! ξ such that J2 = 1. A CR structure J can extend to Cξ and decomposes Cξ into the direct sum of T1,0 and T0,1 which are eigenspaces of J with respect to eigenvalues i and i, respectively. A pseudohermitian structure compatible with ξ is a CR structure J compatible with ξ together with a choice of contact form θ and ξ = ker θ. Such a choice determines a unique real vector …eld T transverse to ξ, which is called the Reeb vector …eld of θ, such that θ(T) = 1 and LT θ = 0 or dθ(T, ) = 0.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 43 / 55

Pesudohermitian (2n+1) manifold

Let fT, Zα, Z¯

αg be a frame of TM C, where Zα is any local

frame of T1,0, Z¯

α = Zα 2 T0,1 and T is the characteristic

vector …eld. Then fθ, θα, θ¯

αg, which is the coframe dual to

fT, Zα, Z¯

αg, satis…es

dθ = ihαβθα ^ θβ (6.1) for some positive de…nite hermitian matrix of functions (hα ¯

β).

The Levi form h , iLθ is the Hermitian form on T1,0 de…ned by

hZ, WiLθ = i

• dθ, Z ^ W
• .

We can extend h , iLθ to T0,1 The Levi form induces naturally a Hermitian form on the dual bundle of T1,0, denoted by h , iL

θ,

and hence on all the induced tensor bundles.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 44 / 55

Pesudohermitian (2n+1) manifold

The pseudohermitian connection of (J, θ) is the connection r

• n TM C (and extended to tensors) given in terms of a local

frame Zα 2 T1,0 by

rZα = θαβ Zβ, rZ¯

α = θ¯ α ¯ β Z ¯ β,

rT = 0,

where θαβ are the 1-forms uniquely determined by the following equations: dθβ = θα ^ θαβ + θ ^ τβ, 0 = τα ^ θα, 0 = θαβ + θ ¯

β ¯ α,

(6.2) We can write (by Cartan lemma) τα = Aαγθγ with Aαγ = Aγα

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 45 / 55

Pesudohermitian (2n+1) manifold

dθβ

α θβ γ ^ θγα = Rβ αρ¯ σθρ ^ θ ¯ σ + iθβ ^ τα iτβ ^ θα (mod θ)

The pseudohermitian Ricci curvature tensor Ric and the torsion tensor Tor are de…ned on T1,0 by Ric(X, Y ) = Rα ¯

βX αY ¯ β

Tor(X, Y ) = i ∑α,β(A¯

α ¯ βX ¯ αY ¯ β AαβX αY β).

Here X = X αZα , Y = Y βZβ, Rα ¯

β = Rγδ α ¯ β and Rγγ α ¯ β is the

pseudohermitian curvature tensor.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 46 / 55

Pesudohermitian (2n+1) manifold

For a real function u, the subgradient rb is de…ned by rbu 2 ξ and hZ, rbuiLθ = du(Z) for all vector …elds Z tangent to contact plane. Locally rbu = ∑α u¯

αZα + uαZ¯ α. We can use the

connection to de…ne the subhessian as the complex linear map

(rH)2u : T1,0 T0,1 ! T1,0 T0,1

by

(rH)2u(Z) = rZrbu.

In particular,

jrbuj2 = 2uαuα, j(rH)2uj2 = 2(uαβuαβ + uαβuαβ).

Also ∆bu = Tr (rH)2u = ∑α(uα¯

α + u¯ αα).

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 47 / 55

Pesudohermitian (2n+1) manifold

De…nition 6.1

Let (M, J, θ) be a pseudohermitian (2n + 1)-manifold. A piecewise smooth curve γ : [0, 1] ! M is said to be horizontal if γ0(t) 2 ξ whenever γ0(t) exists. The length of γ is then de…ned by l(γ) =

Z 1

0 hγ0(t), γ0(t)i

1 2

Lθdt.

The Carnot-Carathéodory distance between two points p, q 2 M is dc(p, q) = inffl(γ) j γ 2 Cp,qg where Cp,q is the set of all horizontal curves joining p and q.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 48 / 55

Pesudohermitian (2n+1) manifold

By Chow connectivity theorem, there always exists a horizontal curve joining p and q, so the distance is …nite. Furthermore, there is a minimizing geodesic joining p and q so that its length is equal to the distance dc(p, q).

De…nition 6.2

We say M is complete if it is complete as a metric space.

Shu-Cheng ChangDepartment of Mathematics and TIMS, NTU(joint work with Ting-Jung Kuo and Jingzhu Tie) () CR Yau’s Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions 49 / 55

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