Advances in Pure Mathematics, 2011, 1, 5-8
doi:10.4236/apm.2011.12003 Published Online March 2011 (http://www.SciRP.org/journal/apm)
Copyright © 2011 SciRes. APM
The Harmonic Functions on a Complete Asymptotic
Flat Riemannian Manifold*
Huashui Zhan
School of Sciences, Jimei University, Xiamen, China
E-mail: hszhan@jmu.edu.cn
Received January 6, 2011; revised March 9, 2011; accepted March 15, 2011
Abstract
Let
M
be a simply connected complete Riemannian manifold with dimension 3n. Suppose that the sec-
tional curvature satisfies

2
2
1
Ma
bK
 
, where
is distance function from a base point of
M
,
, ab are constants and 0ab
. Then there exist harmonic functions on
M
.
Keywords: Harmonic Function, Riemannian Manifold, Negative Sectional Curvature
1. Introduction
The existence of the harmonic functions on a complete
Riemannian manifold is a well known problem. In what
follows, we consider the harmonic function
f
is not a
constant function, that is, , =
f
cc
constant. If there is
no restrictions imposed on the curvature, then it was
proved [1] that there does not exist a harmonic function
of the form

, 1<<
p
LM p
, on the manifold. If
=p, then it was proved [1] that there dose not exist
any bounded harmonic function on a complete manifold
with nonnegative Ricci curvature. On the other hand, by
introducing the sphere at infinity

S, Anderson-
Scheon [2] and Sullivan [3] succeeded to prove the
existence of the bounded harmonic functions on a com-
plete simply-connected manifold with
22
<0,
M
bK a 
where
M
K
represents the sectional curvature and 0a
,
0b are constants. It is naturally to consider whether
the same conclusion holds only on the manifold with
negative sectional curvature, i.e. 2<0
M
bK ? How-
ever, this is still an open problem.
Let
be a complete manifold and oM be fixed.
Then we write
 
min ,
o
Kc
(1)
if for any minimal geodesic
issuing from o, the
sectional curvature of the plane which is tangent to
is
greater than or equal to

c
, where

c
is a mo-
notone increasing function and
is the distance func-
tion from the base point o in the manifold. This notion
was first introduced by Klingenberg [4]. By using the
Toponogov-type comparison theorem with min
o
K
c in
[5,6], and using the approach of Anderson and Scheon
[2], we are able to prove the following result:
Theorem 1. Let
be a complete simply-connected
Riemannian manifol d with dimension 3n. If
 
2
2min max,
1
MM a
bK K

 
(2)
with
 
1<2,2 121>7,ba nab (3)
then there exist bounded harmonic functions on the
manifold
, where
is distance function from a
given base point o in
, 0ab .
A special case of the manifolds satisfying the theorem
1 is with the following sectional curvature condition
 
22
min max.
11
MM
aa
KK

 
(4)
In general, since
is large enough, the curvature in
(4) is close to 0, one would conjecture that the behavior
of this manifold would be much closer to the Euclidean
spaces and hence there may not exist any bounded har-
monic functions. Our theorem states that this conclusion
is not true.
*The paper is supported by Natural Science Foundation of Fujian
Province in China (No. 2009J01009), and Pan Jinlong SF of Jimei
University in China.
H. S. ZHAN
Copyright © 2011 SciRes. APM
6
2. The proof of Theorem 1
Let

2
M
c be the complete simply connected surface
of constant curvature c. We also assume that all
geodesics have unit speed.
Lemma 2. ([5-7]). Let
be a complete Rieman-
nian manifold and obe a point of
with min
o
K
c.
1) Let
:,, =0,1,2
ii
olMi
be minimal geode-
sics with
 
122 011
0==, 0=lo l
 
and


200
0= l

. Then, there exist minimal geodesics

2
:,, =0,1,2
ii
olMci
with
 
122011
0=, 0=ll
 
 and

200
0= l


such that
 
=, =0,1,2
ii
LLi

and
 

 
11 011 0
,0 ,0,ll
 
 
 





00 200 2
,0 ,0.ll
 
 
 

2) Let
:,, =1,2
ii
olMi
be two minimizing geo-
desics starting from p. Let

2
:,, =0,1,2
ii
olMci
be minimizing geodesics starting from same point such
that
 

 

12 12
0, 00, 0
 
 
 
. Then
 

 
11 2211 22
,,,
c
dl ldl l
 

where c
d denotes the distance function in
2
M
c.
If

max
o
K
c
, then we have the parallel result as
Lemma 2.
Let
be a complete Riemannian manifold, o a
point of
with

max
o
Kc

, and

c
a mo-
notone increasing function. For any given 0>0
, it is
obvious that


max
00
, 0
o
Kc
 
. By Lemma
1 and the hyperbolic cosine theorem in

2
0
Mc
, we
can easily prove the following lemma
Lemma 3. Let
be a complete Riemannian mani-
fold and o a point of
. For any given >0r, let
112
,,oxx be three points in
such that

11 12
=,=,dox dox
. Suppose that (2) is true. De-
note the ray from o to 1
x
by 1
, the ray from o to
2
x
by 2
and the angle of 1
and 2
at o by
;


 
1
12
21 ,
2ln1
2
,2 ln1,
doo
a
dxx b





(3)
where
is large enough and
is small enough.
Now we consider a simply connected Riemannian
manifold
with negative sectional curvature. As usual,
two rays 1
and 2
on
are equivalent, that is,
12

if and only if
 

12
dt tc

, for all 0t.
If we denote the set of all rays in
by , then the
Matrin boundary at infinity is defined as .
If 1
and 2
are emanating from the same point o
of
,
12
0= 0

, from (2),


1
21 ,
2ln1,
doo c
a



as 0t, =0
. This means that 12

if and only
if 12
=

. Then,
==:remanating from a fixed point So
 and
it is equivalent to the unite sphere o
S in o
TM.
Moreover, by Lemma 3, we can also construct a C
topological structure on

=MMS
as [1]. By
using this fact, we can prove the following Theorem 4,
and Theorem 1 as its Corollary.
Theorem 4. Let M be a simply connected Riemannian
manifold with (2) and (3). For any

0
CS

, there
is a unique harmonic function

0
uC MCM

such that

=
S
u
.
Proof: We first fix the base point o. Let
S
be
equivalent to the unit tangent sphere

1
1= n
o
SS
. From
[1], without loss of generality, we may assume that
1
o
CS
. Since 0
M
K,
is diffeomorphic
to n
R. Denote
 
1
,1=
n
o
rSS

as the normal
coordinate around o. Then,

=, 1
o
S
 
. Now,
we define an extension of
and still denote it by
,
so that
,= ,forall>0.rr
 
Then
is a differential function on

\0M. Write
 
 
11
osc =,
sup
BxyB
x
y
x

Now, we proceed to prove Theorem 4 via the follow-
ing steps:
1)

1
1
1
osc= e
BxO




. According to the defi-
nition of
, if
1
x
yB, then

=,yx c
 



where ,
is the geodesic sphere coordinate of , yx,
respectively. By Lemma 3, we have

221ln11,
x
xy


1
1
e.c


2) Consider

such that
1
1
=eO




.
Let
0
CR
, 10

,
sup 1,1p
 . Set
H. S. ZHAN
Copyright © 2011 SciRes. APM
7





2
2
d
=.
d
x
M
x
M
yyy
yy


Then



 




 

2
1
2
1
1
(1)
d
=d
= osc.
sup
x
Bx
x
Bx
Bx
Bx
yyxy
yy
yx
 
 


At the same time, we have










00
=0
2
0
2=0
=
=d.
d
xx
y
Mxx
y
M
xxx
xxxy
xx

 








Now it is not difficult to show that (c.f.[1,2])

1
osc .
Bx
xc

3) Consider the function
 
1
12
=e ,δ==1
1
cc
gCx c

. Then,

3
2
δ=1 ,
2
c
 
 

53
2
22
53
22
3
δ=1 1
22
3
=11.
22
c
c




 

 


The last equality is due to 21
. Hence, we
deduce the followings:

 
δ
=e δδ ,
xx
g
xx xx

 

 

 

   
 
2
δδ
2
δ2
5331
312
δ2δ222 2
=e δδ eδ2δδ
= eδδ2δδ δ2δδ
131
= e111e1111
442
xx xx
xx
gxxxx xxxxxx
xx xxxx xxxx xxx
cc


 

 






 

 
 

 


.
(4)
For any fixed point pM, denote
0=,dop
, and denote
=,dox
for any

0
,xBo
. Then
 
2
2min max
0
,
1
MM a
bK K

 
(5)
which means that


00 0
111 1
coth 1,
11 1
nanaannb




  (6)
by (4), we have
 


 
 
 




2δ
32
δ
533 1
00
222 2
22 2
δ
32 533
22 2
1
e1
41 1
111 1
311
ecoth
11
41 1211
11 1
31
e11
41 141 121
gc
nb naan
c
nbna
cc c
ce




 
 

  


 





 




 


  






 



 






22
δ
25533
222 2
2
δ
311 11
13
= e1
41 11
41 41 41121
72 1141
e< 0
141
cnbcnac
cccc
nbnac
c
 
 

 
 
 
 


 
 
  
 
 







H. S. ZHAN
Copyright © 2011 SciRes. APM
8
provided that c is small enough and by the conditions

1<2,2 121>7.banab
Hence,
<0.g
It is obvious that
 
δ11
ee
1
cx
xx
so that
there exists a constant 1
c such that

1.cg

According to the well-known Perron canonical har-
monic function theorem, the barrier functions cg
and cg
assure that there exists a harmonic function
u satisfying
11
.cg ucg


Now, it is easy to verify that u satisfies the boundary
conditions. Thus, Theorem 4 is proved.
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[3] D. Sullivan, “The Ditchless Problem at Infinity for a
Negatively Curved Manifold,” Journal of Differential
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[4] W. Klingenberg, “Riemannian Geometry,” De Grunter,
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[5] Y. Macrognathia, “Manifolds with Pinched Radial Cur-
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[7] C. Y. Xia, “Open Manifolds with Nonnegative Ricci
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