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AdvancesinAppliedMathematics
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,2021,10(11),3912-3922
PublishedOnlineNovemb er2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.1011416
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Liouville
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GradientEstimateofSubelliptic
HarmonicMapsonSub-Riemannnian
Manifolds
WentingZou
CollegeofMathematicsandComputerScience,ZhejiangNormalUniversity,
JinhuaZhejiang
Received:Oct.19
th
,2021;accepted:Nov.9
th
,2021;published:Nov.22
nd
,2021
Abstract
TheRiemannianfoliationisaspecialclassofsub-Riemannianmanifolds,ifitshori-
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x
zontaldistributionsatisfiesthebracketgeneratingcondition.Inthispaper,westudy
thegradientestimationofthesubellipticharmonicmapsandtheLiouville-type
theorems.
Keywords
Sub-RiemannianManifolds,SubellipticHarmonicMaps,RiemannianFoliation,
GradientEstimate,Liouville’sTheorem
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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cos
(
√
κt
)
κ
,κ>
0
t
2
2
,κ
= 0
(2.20)
Ú
ψ
(
q
) =
φ
◦
ρ
(
q
)(2.21)
d
i
ù
A
Û
¥
Hessian
'
½
n
Œ
•
Hessψ
≥
cos
(
√
κρ
)
·
h
(2.22)
Ú
n
3
(
ë
„
[7])
e
0
<D<
π
2
√
κ
,
K
•
3
ν
∈
[1
,
2)
,
b>φ
(
D
)
Ú
δ>
0
=
•
6
u
D
,
¦
ν
cos(
√
κt
)
b
−
φ
(
t
)
−
2
κ
≥
δ,
∀
t
∈
[0
,D
](2.23)
A
O
/
,
X
J
f
´
l
ÿ
/
i
ù
“
G
(
M
¡
-
Ç
k
š
K
þ
.
κ
i
ù6
/
N
K
¥
B
D
(
y
0
)
þ
g
ý
N
Ú
N
.
@
o
ν
∆
H
(
ψ
◦
f
)
b
−
ψ
◦
f
−
2
κ
|
d
H
f
|
2
≥
δ
|
d
H
f
|
2
(2.24)
3.
Y
²
F
Ý
O
3
!
m
©
§
·
‚
k
‰
Ñ
˜
‡
A
Ï
¼
ê
½
Â
:
½
Â
2
X
J
M
þ
•
3
˜
‡
1
w
¼
ê
r
:
M
→
[0
,
+
∞
)
´
•
;
,
…
•
3
~
ê
C
3
,
¦
|∇
H
r
|≤
C
3
,
∆
H
r
≥−
C
3
1+
1
r
(3.1)
K
¡
r
÷
v
'
½
n
5
Ÿ
.
5
1
e
M
´
Ricci
-
Ç
k
e
.
i
ù6
/
,
K
§
å
l
¼
ê
÷
v
(3.1)
.
(
M
m
+
d
,H,g
H
;
g
)
´
˜
‡
š
;
g
i
ù6
/
,
3
B
2
R
(
x
0
)
⊂
M
þ
÷
v
Ric
H
≥−
k
…
|
T
|
,
|
div
H
T
|≤
k
1
(3.2)
DOI:10.12677/aam.2021.10114163917
A^
ê
Æ
?
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q
©
x
Ù
¥
k
,
k
1
≥
0.
r
´
M
þ
÷
v
'
½
n
5
Ÿ
¼
ê
,
…
¼
ê
ϕ
∈
C
∞
0
([0
,
∞
)),
¦
ϕ
|
[0
,
1]
= 1
,ϕ
|
[2
,
∞
]
= 0
,
−
C
0
4
|
ϕ
|
1
2
≤
ϕ
≤
0(3.3)
Ù
¥
C
0
4
´
~
ê
.
-
χ
(
r
) =
ϕ
(
r
R
),
K
|∇
H
χ
|
2
χ
≤
C
4
R
2
,
…
∆
H
χ
≥−
C
4
R
3
B
2
R
\
Cut
(
x
0
)
þ
¤
á
(3.4)
Ù
¥
C
4
=
C
4
(
m,k,k
1
).
•
O
|
d
H
f
|
2
,
•
Ä
9
Ï
¼
ê
Φ
µχ
=
|
d
H
f
|
2
+
µχ
|
d
V
f
|
2
(3.5)
Ù
¥
µ
•
–
½
ê
.
Ú
n
4
3
x
∈
B
2
R
(
x
0
)
?
,
e
χ
(
x
)
6
= 0
,
K
k
∆
H
Φ
µχ
≥
1
2
−
ε
1
|∇
H
Φ
µχ
|
2
Φ
µχ
−
2
κ
|
d
H
f
|
2
Φ
µχ
+
1
2
ε
1
η
min
+
µ
∆
H
χ
−
3
ε
−
1
1
µχ
−
1
|∇
H
χ
|
2
|
d
V
f
|
2
−
2
k
+
C
2
2
+
C
2
2
ε
1
η
min
|
d
H
f
|
2
(3.6)
yyy
²²²
±
e
O
Ž
Ñ
3
x
:
?
?
1
,
-
ε
2
=
ε
1
µχ
C
2
,
d
(2.17)
Ú
(2.18)
,
∆
H
Φ
µχ
=∆
H
|
d
H
f
|
2
+
µχ
|
d
V
f
|
2
≥
(2
−
2
ε
1
)
|∇
H
d
H
f
|
2
−
2
k
+
C
2
2
+
C
2
2
ε
1
η
min
|
d
H
f
|
2
+
1
2
ε
1
η
min
|
d
V
f
|
2
−
2
C
2
ε
2
|∇
H
d
V
f
|
2
−
2
κ
|
d
H
f
|
4
+
µ
∆
H
χ
|
d
V
f
|
2
+4
µ
h∇
H
χ
⊗
d
V
f,
∇
H
d
V
f
i
+2
µχ
|∇
H
d
V
f
|
2
−
2
κµχ
|
d
H
f
|
2
|
d
V
f
|
2
=(2
−
2
ε
1
)
|∇
H
d
H
f
|
2
+
µχ
|∇
H
d
V
f
|
2
+4
µ
h∇
H
χ
⊗
d
V
f,
∇
H
d
V
f
i−
2
κ
Φ
µχ
|
d
H
f
|
2
+
1
2
ε
1
η
min
+
µ
∆
H
χ
|
d
V
f
|
2
−
2
k
+
C
2
2
+
C
2
2
ε
1
η
min
|
d
H
f
|
2
(3.7)
DOI:10.12677/aam.2021.10114163918
A^
ê
Æ
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Ð
q
©
x
d
Cauchy
Ø
ª
,
K
k
e
O
:
|∇
H
Φ
µχ
|
2
=
∇
H
|
d
H
f
|
2
+
µχ
|
d
V
f
|
2
2
≤
4
|
d
H
f
+
√
µχd
V
f
|
2
·
∇
H
d
H
f
+
√
µχ
∇
H
d
V
f
+
√
µ
∇
H
χ
2
√
χ
⊗
d
V
f
2
=4Φ
µχ
|∇
H
d
H
f
|
2
+
µχ
|∇
H
d
V
f
|
2
+
µ
|∇
H
χ
|
2
4
χ
|
d
V
f
|
2
+
µ
h∇
H
d
V
f,
∇
H
χ
⊗
d
V
f
i
!
l
,
(2
−
2
ε
1
)
|∇
H
d
H
f
|
2
+
µχ
|∇
H
d
V
f
|
2
+4
µ
h∇
H
χ
⊗
d
V
f,
∇
H
d
V
f
i
=(2
−
4
ε
1
)
|∇
H
d
H
f
|
2
+
µχ
|∇
H
d
V
f
|
2
+2
ε
1
µχ
|∇
H
d
V
f
|
2
+4
µ
h∇
H
χ
⊗
d
V
f,
∇
H
d
V
f
i
≥
1
2
−
ε
1
|∇
H
Φ
µχ
|
2
Φ
µχ
−
1
2
−
ε
1
µ
|∇
H
χ
|
2
χ
|
d
V
f
|
2
+(2+4
ε
1
)
µ
h∇
H
d
V
f,
∇
H
χ
⊗
d
V
f
i
+2
ε
1
µχ
|∇
H
d
V
f
|
2
≥
1
2
−
ε
1
|∇
H
Φ
µχ
|
2
Φ
µχ
−
1
2
−
ε
1
µ
|∇
H
χ
|
2
χ
|
d
V
f
|
2
−
(2+4
ε
1
)
2
2
µ
|∇
H
χ
|
2
4
χ
|
d
V
f
|
2
≥
1
2
−
ε
1
|∇
H
Φ
µχ
|
2
Φ
µχ
−
3
ε
−
1
1
µ
|∇
H
χ
|
2
χ
|
d
V
f
|
2
(3.8)
ò
(3.8)
“
\
(3.7)
=
Œ
¤
y
²
.
e
5
´
½
n
1
y
²
:
y
²
Š
â
Ú
n
3,
Œ
·
ν
∈
[1
,
2),
b>φ
(
D
)
÷
v
(2.23).
-
F
µχ
=
Φ
µχ
(
b
−
ψ
◦
f
)
ν
.
Ù
¥
ψ
d
(2.21)
½
Â
.
x
´
B
2
R
(
x
0
)
þ
χF
µχ
˜
‡
š
"
4
Œ
Š
:
,
K3
x
:
?
,
k
0 =
∇
H
ln(
χF
µχ
) =
∇
H
χ
χ
+
∇
H
Φ
µχ
Φ
µχ
+
ν
∇
H
(
ψ
◦
f
)
b
−
ψ
◦
f
(3.9)
0
≥
∆
H
ln(
χF
µχ
) =
∆
H
χ
χ
−
|∇
H
χ
|
2
χ
2
+
∆
H
Φ
µχ
Φ
µχ
−
|∇
H
Φ
µχ
|
2
Φ
2
µχ
+
ν
∆
H
(
ψ
◦
f
)
b
−
ψ
◦
f
+
ν
|∇
H
(
ψ
◦
f
)
|
2
(
b
−
ψ
◦
f
)
2
(3.10)
DOI:10.12677/aam.2021.10114163919
A^
ê
Æ
?
Ð
q
©
x
|
^
Ú
n
4,(3.10)
3
x
:
?
Œ
z
•
0
≥
∆
H
χ
χ
−
|∇
H
χ
|
2
χ
2
−
1
2
+
ε
1
|∇
H
Φ
µχ
|
2
Φ
2
µχ
−
2
κ
|
d
H
f
|
2
+
ν
∆
H
(
ψ
◦
f
)
b
−
ψ
◦
f
+
ν
|∇
H
(
ψ
◦
f
)
|
2
(
b
−
ψ
◦
f
)
2
+
1
2
ε
1
η
min
+
µ
∆
H
χ
−
3
ε
−
1
1
µχ
−
1
|∇
H
χ
|
2
|
d
V
f
|
2
Φ
µχ
−
2
k
+
C
2
2
+
C
2
2
ε
1
η
min
|
d
H
f
|
2
Φ
µχ
ε
1
=
1
2
ν
−
1
4
,
d
(3.9)
ª
−
1
2
+
ε
1
|∇
H
Φ
µχ
|
2
Φ
2
µχ
=
−
1
2
+
ε
1
−
∇
H
χ
χ
−
ν
∇
H
(
ψ
◦
f
)
b
−
ψ
◦
f
2
≥−
1
2
+
ε
1
(1+
ε
−
1
3
)
|∇
H
χ
|
2
χ
2
−
1
2
+
ε
1
(1+
ε
3
)
v
2
|∇
H
(
ψ
◦
f
)
|
2
(
b
−
ψ
◦
f
)
2
-
ε
3
=
2
−
ν
2+
ν
>
0,
K
1
2
+
ε
1
(1+
ε
−
1
3
) =
2+
ν
ν
(2
−
ν
)
,
1
2
+
ε
1
(1+
ε
3
)
ν
2
=
ν
.
u
´
0
≥
∆
H
χ
χ
−
(1+
2+
ν
ν
(2
−
ν
)
)
|∇
H
χ
|
2
χ
2
+
ν
4
H
(
ψ
◦
f
)
b
−
ψ
◦
f
−
2
κ
|
d
H
f
|
2
+(
1
2
ε
1
η
min
+
µ
∆
H
χ
−
3
ε
−
1
1
µχ
−
1
|∇
H
χ
|
2
)
|
d
V
f
|
2
Φ
µχ
−
2
k
+
C
2
2
+
C
2
2
ε
1
η
min
|
d
H
f
|
2
Φ
µχ
d
(3.4)
†
Ú
n
3,
0
≥−
C
4
χR
−
(1+
2+
ν
ν
(2
−
ν
)
)
C
4
χR
2
+
δ
|
d
H
f
|
2
+(
1
2
ε
1
η
min
+
µ
∆
H
χ
−
3
ε
−
1
1
µ
C
4
R
2
)
|
d
V
f
|
2
Φ
µχ
−
2
k
+
C
2
2
+
C
2
2
ε
1
η
min
|
d
H
f
|
2
Φ
µχ
≥−
C
ν
χR
+
δ
|
d
H
f
|
2
+(
1
2
ε
1
η
min
−
µC
ν
R
)
|
d
V
f
|
2
Φ
µχ
−
2
k
+
C
2
2
+
C
2
2
ε
1
η
min
|
d
H
f
|
2
Φ
µχ
(3.11)
Ù
¥
C
ν
=
C
ν
(
ν,C
4
),
δ>
0
d
Ú
n
3
û
½
…
=
•
6
u
D
.
qd
Φ
µχ
=
|
d
H
f
|
2
+
µχ
|
d
V
f
|
2
,
|
d
V
f
|
2
=
µ
−
1
χ
−
1
(Φ
µχ
−|
d
H
f
|
2
)
DOI:10.12677/aam.2021.10114163920
A^
ê
Æ
?
Ð
q
©
x
¤
±
(3.11)
Œ
z
•
0
≥
−
C
ν
χR
+
δ
|
d
H
f
|
2
+(
1
2
ε
1
η
min
−
µC
ν
R
)
µ
−
1
χ
−
1
+(
−
1
2
µ
−
1
χ
−
1
ε
1
η
min
+
χ
−
1
C
ν
R
−
2
k
−
C
2
2
−
C
2
2
ε
1
η
min
)
|
d
H
f
|
2
Φ
µχ
≥
1
χ
(
1
2
ε
1
µ
−
1
η
min
−
2
C
ν
R
)
+[
δχ
Φ
µχ
−
1
2
ε
1
µ
−
1
η
min
+2
k
+
C
2
2
ε
1
µ
+
C
2
2
ε
1
η
min
]
|
d
H
f
|
2
χ
Φ
µχ
(3.12)
•
¦
1
2
ε
1
µ
−
1
η
min
−
2
C
ν
R
š
K
§
·
‚
Œ
±
ε
1
µ
−
1
=
4
C
ν
η
min
R
,
Q
µ
−
1
=
4
C
ν
ε
1
η
min
R
.
u
´
χ
Φ
µχ
≤
δ
−
1
4
C
ν
R
+2
k
+
C
2
2
ε
1
µ
+
C
2
2
ε
1
η
min
(3.13)
=
δ
−
1
C
5
Ù
¥
C
5
=
4
C
ν
R
+2
k
+
C
2
2
ε
1
µ
+
C
2
2
ε
1
η
min
.
Ï
d
max
B
2
R
(
x
0
)
χF
µχ
≤
χ
Φ
µχ
(
b
−
ψ
◦
f
)
ν
(
x
)
≤
C
5
δ
(
b
−
φ
(
D
))
ν
(3.14)
d
d
max
B
R
(
x
0
)
|
d
H
f
|
2
≤
b
ν
max
B
R
(
x
0
)
F
µχ
≤
C
5
b
ν
δ
(
b
−
φ
(
D
))
ν
(3.15)
K
max
B
R
(
x
0
)
|
d
H
f
|
2
≤
4
C
ν
R
+2
k
+
C
2
2
ε
1
µ
+
C
2
2
ε
1
η
min
=
C
1
k
+
1
R
(3.16)
Ù
¥
C
1
•
•
6
u
k
,
k
1
,
κ
,
D
.
l
¤
½
n
1
y
²
.
ë
•
©
z
[1]Yau,S.(1975)HarmonicFunctionsonCompleteRiemannianManifolds.
Communicationson
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