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Mathematics 2019, 7(4), 351; https://doi.org/10.3390/math7040351
Some Liouville Theorems on Finsler Manifolds
Department of Mathematics and Physics, Hefei University, Hefei 230601, China
Department of Mathematics and Computer Science, Tongling University, Tongling 244000, China
Key Laboratory of Applied Mathematics (Putian University), Fujian Province University, Putian 351100, China
Author to whom correspondence should be addressed.
Received: 22 January 2019 / Accepted: 11 April 2019 / Published: 15 April 2019
We give some Liouville type theorems of harmonic (resp. subharmonic, superharmonic) functions on a complete noncompact Finsler manifold. Using the geometric relationship between a Finsler metric and its reverse metric, we remove some restrictions on the reversibility. These improve the recent literature (Zhang and Xia, 2014).
Keywords:harmonic function; Finsler manifold; Liouville theorem; reversibility
MSC:Primary 53C60; Secondary 53B40
As is well known, Liouville theorems play an important role in analyzing the underlying manifolds. The classical Liouville theorem states that any nonnegative (or bounded) harmonic function on must be constant. Up to now, there are many generalizations studied on complete Riemannian manifolds. Yau [1,2] proved that any positive (or bounded) harmonic function on a complete Riemannian manifold with nonnegative Ricci curvature must be constant and there are no nonnegative subharmonic functions on such a Riemannian manifold for . Yau’s results were then generalized by Sturm and Schoen, etc. See [3,4,5] and references therein. For general symmetric diffusion operators, Li  extended various Liouville theorems as above.
Recently, Zhang-Xia , Yin-He  and Yin-Zhang  extended the above Liouville theorems in the Finsler setting. Notice that, in [7,8], the Finsler manifolds discussed must have finite reversibility. In this paper, we show that this restriction can be removed. Specifically, we obtain the following results.
Let be an n-dimensional forward complete noncompact Finsler manifold. If a positive function on M satisfies on andwhere , then u is a constant. In particular, if and on , then u is a constant.
Let be an n-dimensional complete noncompact Finsler manifold. Assume that
- If and is a nonnegative superharmonic function on M, then u is a constant.
- If and is a nonnegative subharmonic function on M, then u is a constant.
Here, is defined in (2) below, and some important concepts such as Finsler metric, Finsler Laplacian and harmonic (resp. subharmonic, superharmonic) functions will be given in Section 2, respectively.
If the Finsler manifold is compact, then, by the divergence theorem, we can prove all harmonic (resp. subharmonic, superharmonic) functions are constant. Theorem 1 can be regarded as a generalization of Theorem 1 in  when . If is a Riemannian metric measure space, then Theorem 2 is exactly Theorem 1 in  or Theorem 13.1 in .
In comparison with , the condition on the reversibility is deleted in theorems above. There are many Finsler manifolds with infinity reversibility. Consider the Randers metric inwhere denotes the standard Euclid norm. Then the reversibility
The geometric quantities between F and its reverse metric have many important relationships. For example,
- A forward (backward) distance function w.r.t. F is a backward (forward) distance function w.r.t. , and vice versa.
- A forward (backward) geodesic ball w.r.t. F is a backward (forward) geodesic ball w.r.t. , and vice versa.
- If f is a superharmonic (subharmonic) function w.r.t. Δ, then is a subharmonic (superharmonic) function w.r.t. , and vice versa.
Let M be an dimensional smooth manifold and be the natural projection from the tangent bundle . Let be a point of with , , and let be the local coordinates on with . A Finsler metric on M is a function satisfying the following properties:
- Regularity: is smooth in ;
- Positive homogeneity: for ;
- Strong convexity: The fundamental quadratic form
Let be a vector field. Then, the covariant derivative of X by with reference vector is defined bywhere denote the coefficients of the Chern connection.
For a smooth function u, the gradient vector of u iswhere is Legendre transformation defined as
Let be a smooth vector field on M. The divergence of V with respect to an arbitrary smooth volume form is defined bywhere . Then, the Finsler Laplacian of u can be defined by
Since is undefined at x where , the definition can be viewed in distributional sense. That is, for ,
We note here that since the gradient operator ∇ is not linear operator in general, the Finsler Laplacian is quite a bit different from the Riemannian Laplacian. Given a vector field V such that on , where , the weighted gradient vector and the weighted Laplacian on the weighted Riemannian manifold are defined by
It follows that
Let u be a positive harmonic function on M, . It was proved that (see ). We say that is a subharmonic (resp. superharmonic) function on M if (resp. ). In a weak sense, u is a subharmonic (resp. superharmonic) function in M if, for any positive function , it holds
Let be a Finsler n-manifold. Fix a point . We denote a forward (resp. backward) geodesic ball of radius r with center at by (resp.).
Let be a Finsler n-manifold and . Then, there exists a function defined bysuch thatwhere C is a positive constant.
Let be a smooth function on the real line with and such that
Clearly, , where C is some positive constant. Definewhere is the distance function form . Then,□
Notice that and it is differentiable almost everywhere on with bounded differential. Since a subharmonic (resp. superharmonic) function u belongs to , and a.e. on (Lemma 3.5 in ), we find the Formula (1) still holds for this .
3. Proof of the Main Theorems
For any nonnegative function u, set
Note that if .
Proof of Theorem 1.
Set . Then, in , one obtainswhich gives
Let be the function defined in Lemma 1. Then, it is differentiable almost everywhere on with a bounded differential. Note that a.e. on from Lemma 3.5 in . Thus, by the divergence theorem, we have
Therefore,which implies that
By using Lemma 1 and the definition of , we deduce
Letting , it follows from that everywhere. Since M is connected, v is a constant on M and so is u. □
Proof of Theorem 2.
Without loss of generality, we might as well assume . Otherwise, we can replace it by for some positive number . We first prove (2) in Theorem 2. Let be a fixed point in M and be a number with . Definewith satisfying
Using the divergence theorem and similar arguments above, we have
Set . Then, (4) becomes
From the conditions in Theorem 2 and (3), it follows that
Then, by similar arguments as in , we can also reach
For fixed , taking , we have
Letting , we havewhich means that
Therefore, by arbitrariness of , we conclude that v must be constant on M and so is u.
Now, we are to prove (1) according to the cases , and , respectively.
Case I: .
Let and be as above. Definewith satisfying
By similar arguments, we also obtain (5) for the backward geodesic ball. The remainder of the proof is the same as above.
Case II: .
Set . Then,
Let be a function as in Case I. We can also obtain
Therefore, we have
Then, by the same argument as above, one obtains that u is constant.
Case III: .
For every , set
Then, is a nonnegative superharmonic function in a weak sense. We will prove this by following the arguments in (, p. 178). Let be a symmetric, concave and bounded smooth function with and (think of a smooth approximation of ). Define(think of a smooth approximation of ). Then, . Notice that . By Legendre transformation, we have . Since a.e. on , for defined in Case I, we have
The last step holds because is differentiable almost everywhere on with bounded differential and u is superharmonic. Thus, the claim follows by approximation.
It is shown that a.e. on , on and w.r.t. to the measure (see ). Thus, a.e. on and on . Notice that is differentiable almost everywhere on with a bounded differential. Therefore, by similar arguments, we can also obtain (4) for on . Set for any . Then, we have (5) as follows:
On the other hand, note thatwhich implies that
Then, by the same discussion in the proof of (2) and Case I of (1), we show that this is constant. Since k is arbitrary, the function u is also constant. □
Using Theorem 2, we can reach the following corollaries which extend Theorem 3 in  and Corollary 1 in , respectively.
Let and u be as in Theorem 2.
- If , then every nonnegative superharmonic function is a constant. In particular, if , then every nonnegative superharmonic function on M is a constant.
- If , then every nonnegative subharmonic function is a constant.
Let and u be as in Theorem 2 and u be a nonnegative superharmonic function. If, for a sequence ,then u is a constant, where C is a positive constant.
Y.S. suggeated to study this topic. W.M. checked the calculations and polished the draft.
This project is supported by EYTVSP (No.gxfx2017095), FMDEP (No.2018xs03), AHNSF (No.160808 5MA03), KLAMFJPU (No. SX201805) and TLXYXM (No. 2018tlxyzd02).
The authors like to sincerely thank the Academic Editor for very valuable and helpful comments and suggestions.
Conflicts of Interest
The authors declare no conflict of interest.
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