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AdvancesinAppliedMathematicsA^êÆ?Ð,2021,10(11),3912-3922
PublishedOnlineNovemb er2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.1011416
giù6/þgýNÚNFÝO
qqq©©©xxx
úô“‰ŒÆ§êƆOŽÅ‰ÆÆ§úô7u
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giù6/§gýNÚN§iù“G(§FÝO§Liouville½n
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-
©ÙÚ^:q©x.giù6/þgýNÚNFÝO[J].A^êÆ?Ð,2021,10(11):3912-3922.
DOI:10.12677/aam.2021.1011416
q©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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DOI:10.12677/aam.2021.10114163915A^êÆ?Ð
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DOI:10.12677/aam.2021.10114163916A^êÆ?Ð
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DOI:10.12677/aam.2021.10114163917A^êÆ?Ð
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DOI:10.12677/aam.2021.10114163918A^êÆ?Ð
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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
4χ
|d
V
f|
2
+µh∇
H
d
V
f,∇
H
χ⊗d
V
fi
!
l,
(2−2ε
1
)

|∇
H
d
H
f|
2
+µχ|∇
H
d
V
f|
2

+4µh∇
H
χ⊗d
V
f,∇
H
d
V
fi
=(2−4ε
1
)

|∇
H
d
H
f|
2
+µχ|∇
H
d
V
f|
2

+2ε
1
µχ|∇
H
d
V
f|
2
+4µh∇
H
χ⊗d
V
f,∇
H
d
V
fi
≥

1
2
−ε
1

|∇
H
Φ
µχ
|
2
Φ
µχ
−

1
2
−ε
1

µ|∇
H
χ|
2
χ
|d
V
f|
2
+(2+4ε
1
)µh∇
H
d
V
f,∇
H
χ⊗d
V
fi+2ε
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+4ε
1
)
2
2
µ
|∇
H
χ|
2
4χ
|d
V
f|
2
≥

1
2
−ε
1

|∇
H
Φ
µχ
|
2
Φ
µχ
−3ε
−1
1
µ
|∇
H
χ|
2
χ
|d
V
f|
2
(3.8)
ò(3.8)“\(3.7)=Œ¤y².
e5´½n1y²:
y²ŠâÚn3,Œ·ν∈[1,2),b>φ(D)÷v(2.23).-F
µχ
=
Φ
µχ
(b−ψ◦f)
ν
.Ù¥ψ
d(2.21)½Â.x´B
2R
(x
0
)þχF
µχ
˜‡š"4ŒŠ:,K3x:?,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.2021.10114163919A^êÆ?Ð
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|^Ún4,(3.10)3x:?Œz•
0 ≥
∆
H
χ
χ
−
|∇
H
χ|
2
χ
2
−

1
2
+ε
1

|∇
H
Φ
µχ
|
2
Φ
2
µχ
−2κ|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−3ε
−1
1
µχ
−1
|∇
H
χ|
2

|d
V
f|
2
Φ
µχ
−

2k+
C
2

2
+
C
2
2
ε
1
η
min

|d
H
f|
2
Φ
µχ
ε
1
=
1
2ν
−
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
−2κ|d
H
f|
2
+(
1
2
ε
1
η
min
+µ∆
H
χ−3ε
−1
1
µχ
−1
|∇
H
χ|
2
)
|d
V
f|
2
Φ
µχ
−

2k+
C
2

2
+
C
2
2
ε
1
η
min

|d
H
f|
2
Φ
µχ
d(3.4)†Ún3,
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−3ε
−1
1
µ
C
4
R
2
)
|d
V
f|
2
Φ
µχ
−

2k+
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
−

2k+
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>0dÚn3û½…=•6uD.
qdΦ
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.2021.10114163920A^êÆ?Ð
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¤±(3.11)Œz•
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
−2k−
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
−
2C
ν
R
)
+[δχΦ
µχ
−

1
2
ε
1
µ
−1
η
min
+2k+
C
2
2
ε
1
µ
+
C
2
2
ε
1
η
min

]
|d
H
f|
2
χΦ
µχ
(3.12)
•¦
1
2
ε
1
µ
−1
η
min
−
2C
ν
R
šK§·‚Œ±ε
1
µ
−1
=
4C
ν
η
min
R
,Qµ
−1
=
4C
ν
ε
1
η
min
R
.u´
χΦ
µχ
≤δ
−1

4C
ν
R
+2k+
C
2
2
ε
1
µ
+
C
2
2
ε
1
η
min

(3.13)
=δ
−1
C
5
Ù¥C
5
=
4C
ν
R
+2k+
C
2
2
ε
1
µ
+
C
2
2
ε
1
η
min
.Ïd
max
B
2R
(x
0
)
χF
µχ
≤
χΦ
µχ
(b−ψ◦f)
ν
(x) ≤
C
5
δ(b−φ(D))
ν
(3.14)
dd
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
≤
4C
ν
R
+2k+
C
2
2
ε
1
µ
+
C
2
2
ε
1
η
min
= C
1

k+
1
R

(3.16)
Ù¥C
1
••6uk,k
1
,κ,D.l¤½n1y².
ë•©z
[1]Yau,S.(1975)HarmonicFunctionsonCompleteRiemannianManifolds.Communicationson
PureandAppliedMathematics,28,201-228.https://doi.org/10.1002/cpa.3160280203
[2]Cheng,S.Y.(1980)LiouvilleTheoremforHarmonicMaps.ProceedingsofSymposiainPure
Mathematics,36,147-151.https://doi.org/10.1090/pspum/036/573431
[3]Choi,H.I.(1982)OntheLiouvilleTheoremforHarmonicMaps.ProceedingsoftheAmerican
MathematicalSociety,85,91-94.https://doi.org/10.1090/S0002-9939-1982-0647905-3
DOI:10.12677/aam.2021.10114163921A^êÆ?Ð
q©x
[4]Strichartz, R.S.(1986) Sub-Riemannian Geometry.JournalofDifferentialGeometry,24, 221-
263.https://doi.org/10.4310/jdg/1214440436
[5]Dong,Y.(2021)Eells-SampsonTypeTheoremsforSubellipticHarmonicMapsfromSub-
RiemannianManifolds.JournalofGeometricAnalysis,31,3608-3655.
https://doi.org/10.1007/s12220-020-00408-z
[6]Baudoin,F.(2016)Sub-LaplaciansandHypoellipticOperatorsonTotallyGeodesicRieman-
nianFoliations.Geometry,AnalysisandDynamicsonSub-RiemannianManifolds,259-321.
https://doi.org/10.4171/162-1/3
[7]Chong,T.,Dong,Y.X.,Ren,Y.B.,etal.(2020)Pseudo-HarmonicMapsfromCompleteNon-
compactPseudo-HermitianManifoldstoRegularBalls.JournalofGeometricAnalysis,30,
3512-3541.https://doi.org/10.1007/s12220-019-00206-2
DOI:10.12677/aam.2021.10114163922A^êÆ?Ð

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