The Harmonic Functions on a Complete Asymptotic Flat Riemannian Manifold *

Let M be a simply connected complete Riemannian manifold with dimension 3 n  . Suppose that the sectional curvature satisfies   2 2 1 M a b K        , where  is distance function from a base point of M , , a b are constants and 0 ab  . Then there exist harmonic functions on M .


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 c c  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 L M 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 complete simply-connected manifold with where M K represents the sectional curvature and 0 a  , 0 b  are constants.It is naturally to consider whether the same conclusion holds only on the manifold with negative sectional curvature, i.e.Let M be a complete manifold and o M  be fixed.Then we write 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 monotone increasing function and  is the distance function 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 M be a complete simply-connected Riemannian manifold with dimension with then there exist bounded harmonic functions on the manifold M , where  is distance function from a given base point o in M , 0 ab  .A special case of the manifolds satisfying the theorem 1 is with the following sectional curvature condition 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 harmonic functions.Our theorem states that this conclusion is not true.

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][6][7]).Let M be a complete Riemannian manifold and o be a point of M with Then, there exist minimal geodesics M c  , we can easily prove the following lemma Lemma 3. Let M be a complete Riemannian manifold and o a point of M .For any given > 0 r , let where  is large enough and  is small enough.
Now we consider a simply connected Riemannian manifold M with negative sectional curvature.As usual, two rays 1

 and 2
 on M are equivalent, that is,  , for all 0 t  .
If we denote the set of all rays in M by  , then the Matrin boundary at infinity is defined as   .
If 1  and 2  are emanating from the same point o and it is equivalent to the unite sphere o S in o T M .Moreover, by Lemma 3, we can also construct a C  topological structure on   = M M S   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 Proof: We first fix the base point o .Let   S  be equivalent to the unit tangent sphere From [1], without loss of generality, we may assume that we define an extension of  and still denote it by  , so that Now, we proceed to prove Theorem 4 via the following steps: . According to the defi- where ,    is the geodesic sphere coordinate of , y x , respectively.By Lemma 3, we have The last equality is due to Hence, we deduce the followings: which means that by ( 4), we have provided that c is small enough and by the conditions     Now, it is easy to verify that u satisfies the boundary conditions.Thus, Theorem 4 is proved.
is still an open problem.

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then we have the parallel result as Lemma 2.Let M be a complete Riemannian manifold, o a point of M with  c  a monotone increasing function.For any given 0 > 0

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well-known Perron canonical harmonic function theorem, the barrier functions cg