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).
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 and
where , 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 in
where 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.
To give a more elaborate improvement, we use these relations and thus avoid employing the reversibility. The remainder of the approaches adopted are similar to Zhang-Xia’s paper . See also in [2,5,10] for the Riemannian case.
The contents of the paper are arranged as follows. In Section 2, some fundamental concepts which are necessary for the present paper are given, and some lemmas are contained. In Section 3, we prove the main theorems and give some corollaries.
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
is positive definite.
Let be a vector field. Then, the covariant derivative of X by with reference vector is defined by
where denote the coefficients of the Chern connection.
For a smooth function u, the gradient vector of u is
where 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 by
where . 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 by
where 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. Define
where 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 .
Set . Then, in , one obtains
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
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. □
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 . Define
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 have
which 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. Define
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
In what follows, we will follow the arguments in  (p. 178) with some modifications. Let be a symmetric, convex, and bounded smooth function with and , where is such that . Define
Then, for any positive integer k, it holds that . Moreover, is a superharmonic function in a weak sense. Indeed, by definition, we have , which yields by Legendre transformation. As and thus 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. Moreover, is smooth on the open subset and is also superharmonic, in the classical sense, on . Notice that is differentiable almost everywhere on with bounded differential. Hence, by similar arguments, we can also obtain (4) for on as in case I. Set for any . Then we have (5) as follows:
On the other hand, note that , and thus
which implies that
Then by the same discussion in the proof of (2) and Case I of (1), we show that this is constant. Take then a sequence (such that each satisfies the same properties as ) uniformly converging to the absolute value function. Every is then constant. These constants are bounded (they are in ). Thus, up to pass to a subsequence converges uniformly to and to a constant at the same time. Hence, must be constant. k being arbitrary, 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.
Yau, S.T. Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math.1975, 28, 201–208. [Google Scholar]
Yau, S.T. Some function theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Univ. Math. J.1976, 25, 659–670. [Google Scholar] [CrossRef]
Li, P. Uniqueness of L1 solutions for the Laplace equation and the heat equation on Riemannian manifolds. J. Diff. Geom.1985, 20, 447–457. [Google Scholar] [CrossRef]
Li, P.; Schoen, R. Lp and mean value properties of subharmonic functions on Riemannian manifolds. Acta Math.1984, 153, 279–301. [Google Scholar] [CrossRef]
Sturm, K.T. Analysis on local Dirichlet spaces, I. recurrence, conservativeness and Lp-liouville properties. J. Reine Angew. Math.1994, 456, 173–196. [Google Scholar]
Li, X. Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math. Pures Appl.2005, 84, 1295–1361. [Google Scholar] [CrossRef][Green Version]
Zhang, F.; Xia, Q. Some Liouville-type theorems for harmonic functions on Finsler manifolds. J. Math. Anal. Appl.2014, 417, 979–995. [Google Scholar] [CrossRef]
Yin, S.; He, Q. A generalized Omori-Yau maximum principle in Finsler geometry. Nonlinear Anal. Theory Methods Appl.2015, 128, 227–247. [Google Scholar] [CrossRef]
Yin, S.; Zhang, P. Remarks on Liouville-type theorems on complete noncompact Finsler manifolds. Revista de la Unión Matemática Argentina2018, 59, 255–264. [Google Scholar] [CrossRef]
Grigor’yan, A. Analytic and geometric background of recurrence and nonexplosion of the brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. (N.S.)1999, 36, 135–249. [Google Scholar] [CrossRef]