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Some comparison theorems and their applications in Finsler geometry
Journal of Inequalities and Applications volume 2014, Article number: 107 (2014)
By using arbitrary volume forms, we establish Laplacian comparison theorems for Finsler manifolds under certain curvature conditions. As applications, some volume comparison theorems and Mckean type eigenvalue estimates of Finsler manifolds are obtained. Moreover, we also generalize Calabi-Yau’s linear volume growth theorem, and Milnor’s results on curvature and the fundamental group to the Finsler setting.
MSC: 53C60, 53B40.
In recent years, Finsler geometry has developed rapidly in its global and analytic aspects. The present main work is to generalize and improve some famous theorems of the Riemann geometry to the Finsler setting. Among these issues, the Finsler-Laplacian is one of the most important and interesting projects. As is well known, there are several definitions of the Finsler-Laplacian, including the nonlinear Laplacian, the mean-value Laplacian and so on, in Finsler geometry. With regard to the nonlinear Finsler-Laplacian, some Laplacian comparison theorems, volume comparison theorems, and various estimations on the first eigenvalue have been established [1–5].
In , Shen first generalized comparison theorems to the Finsler geometry. Afterwards, Wu and Xin  proved Laplacian comparison theorems, volume comparison theorems under various flags, and Ricci and S-curvature conditions. Recently, using the Ricci curvature condition, and the distortion τ instead of S-curvature, Wu  and Zhao and Shen  further generalized volume comparison theorems in  and , respectively. It should be noted here that by utilizing the weighted Ricci curvature condition , Ohta and Sturm  and Ohta  gave another version of these theorems, which are more concise than the corresponding ones in  and .
In the Riemannian case, Ding  obtained a new Laplacian comparison theorem by the Ricci curvature condition . Later, this result was generalized to Finsler manifolds in . For a Finsler n-manifold with nonpositive flag curvature, if its Ricci curvature satisfies , then the following holds whenever the distance function ρ is smooth:
In this paper, we shall further improve this theorem by using the weighted Ricci curvature condition and remove the term of the S-curvature. To be precise, we will give the following result.
Theorem 1.1 Let be a Finsler n-manifold with nonpositive flag curvature and nonpositive S-curvature. If the weighted Ricci curvature satisfies , then the following holds whenever the distance function ρ is smooth:
where is defined by (1.1).
In addition to this, the Laplacian comparison theorem under the flag curvature and S-curvature condition is also obtained. As applications, we give some volume comparison theorems under the above-described conditions. It is worth mentioning that all the results we obtained are more concise than those in the related literature [3, 4] and more similar to the Riemannian case in form.
In [8, 9], Calabi and Yau stated that the volume of any complete noncompact Riemannian manifold with nonnegative Ricci curvature has at least linear growth. In , Wu has established the Finsler version of Calabi-Yau’s linear volume growth theorem by using an extreme volume form. His result is
where volmax denotes the volume with respect to the maximal volume form. In the present paper, we will further claim that for an arbitrary volume form Calabi-Yau’s result still holds.
Theorem 1.2 Let be a complete noncompact Finsler n-manifold with finite reversibility λ. If the weighted Ricci curvature satisfies , , then
where (resp. ) denotes the forward (resp. backward) geodesic ball of radius R centered at p and C denotes the constant depending on N, λ, and (resp. ).
In Riemannian geometry, Mckean  proved that if is a complete and simply connected Riemannian n-manifold with sectional curvature , then the first eigenvalue . Afterwards, this result was extended by Ding in , stating that for a complete noncompact and simply connected Cartan-Hadamard manifold satisfying the first eigenvalue can be estimated below by . A few years ago, these results were generalized to the Finsler setting by Wu and Xin . In their paper, some conditions such as ‘finite reversibility’ and some restrictions on S-curvature should be satisfied, which are natural conditions in Finsler geometry and satisfied automatically in the Riemannian case.
In the present paper, we further generalize the Mckean type estimations to Finsler manifolds. We note that our results are as neat and simple as in the Riemannian case.
Theorem 1.3 Let be a complete noncompact and simply connected Finsler n-manifold with finite reversibility λ and nonpositive S-curvature. If the flag curvature satisfies , then
Theorem 1.4 Let be a complete noncompact and simply connected Finsler n-manifold with finite reversibility λ, nonpositive flag curvature and nonpositive S-curvature. If the weighted Ricci curvature satisfies , then
Remark 1.5 In Theorem 1.4, the condition ‘nonpositive flag curvature’ is necessary and it is a substitute for the condition ‘Cartan-Hadamard manifold’ in . Since in the Riemannian case flag curvature is just sectional curvature, this condition is a natural condition.
Remark 1.6 The definitions of the reversibility λ and S-curvature will be given in Sections 2, 4 below. When is a Riemannian manifold, , , and the above two results coincide with  and , respectively. Further, when is a Finsler manifold, the corresponding lower bounds obtained in  are and , respectively.
This paper is organized as follows. In Section 2, the related fundamentals of Finsler geometry such as Finsler metric, weighted Ricci curvature, gradient vector, Finsler-Laplacian, and some lemmas are briefly introduced. The main results will be proved in Sections 3, 4, 5, respectively.
Let M be an n-dimensional smooth manifold and be the natural projection from the tangent bundle TM. Let be a point of TM with , , and let be the local coordinates on TM 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 positively definite.
Let be an open set and be a nonzero vector field on . Define
where are Chern connection coefficients. Then
Given two linearly independent vectors , the flag curvature is defined by
where is the Chern curvature
Then the Ricci curvature for is defined as
where , form an orthonormal basis of with respect to .
For a given volume form and a vector , the distortion of is defined by
To measure the rate of changes of the distortion along geodesics, we define
where is the geodesic with . S is called the S-curvature.
Now we can introduce the weighted Ricci curvature on the Finsler manifolds, which was defined by Ohta in . In the present paper, we reform it as follows.
Definition 2.1 
Let be a Finsler n-manifold. Given a vector , let be a geodesic with , . Define
where denotes the S-curvature at . The weighted Ricci curvature of is defined by
Here we will spend some words about the assumption of the nonpositive S-curvature in this paper. If the Finsler metric F is reversible, then the S-curvature is homogeneous and hence only if . If , then for all N. For instance, the Busemann-Hausdorff measures on Berwald spaces satisfy . Express a Rander metric in terms of a Riemannian metric and a vector by
where and . Set
where is a constant, is an anti-symmetric matrix and is a constant vector. In , we know that F has constant flag curvature and . From , we further get by using the Busemann-Hausdorff measures. For more examples, we can refer to .
For a smooth function u and a smooth vector field V on M, we set and . If on , then the weighted gradient vector of u on the weighted Riemannian manifold is defined by
The divergence of on M with respect to an arbitrary volume form and the Finsler weighted Laplacian of u on are defined by
Let be the Legendre transform. For a smooth function u on M, the gradient vector and the Finsler-Laplacian of u is defined by
In particular, on we have
Let be a differential vector field. Then the covariant derivative of X by with reference vector is defined by
where denotes the coefficients of the Chern connection.
For a smooth vector field V on M and , we define by using the covariant derivative as
We also set for the smooth function u and . Then
Let be a local orthonormal basis with respect to on . Write , then we have
Let be a Finsler manifold. Define the distance function by
where the infimum is taken over all differentiable curves with and .
Lemma 2.2 
Let be a Finsler n-manifold and a smooth function. Then on we have
where and is a local -orthonormal basis on .
Lemma 2.3 
Assume that for . Then the Laplacian of the distance function from any given point can be estimated as follows:
in the sense of distributions on .
Lemma 2.4 
Let be a Finsler n-manifold and be the distance function from a fixed point p. Suppose that the flag curvature of M satisfies . Then for any vector X on M, the following inequality holds whenever ρ is smooth:
where is defined by (1.1).
3 Laplacian comparison theorems
Theorem 3.1 Let be a Finsler n-manifold with nonpositive flag curvature and nonpositive S-curvature. If the weighted Ricci curvature satisfies , then the following holds whenever the distance function ρ is smooth:
Proof Let be the distance function. If ρ is smooth at , then it is also smooth near q. Let be the forward geodesic sphere of radius centered at p. Choosing the local -orthonormal frame of near q, we get local vector fields , by parallel transport along geodesic rays. Using (2.1)-(2.4), we have
Here denotes the Hilbert-Schmidt norm with respect to . We refer to  for details.
Since M has nonpositive flag curvature, from Lemma 2.4 we see that the eigenvalues of are nonnegative. This yields
Note that and for , from (3.1) and (3.2) we have
Notice that and has nonnegative eigenvalues; from Lemma 2.2 we have
for . On the other hand, it is easy to see that since . Combining (3.3) and (3.4), and using Lemma 2.2 again, we obtain
By a simple argument, (3.5) can be rewritten as
Set , , then (3.6) becomes
Since M has nonpositive flag curvature and nonpositive S-curvature, from Lemma 2.4 we get
which implies that there exists such that
From (3.7) we have
which yields , i.e., . □
If M has nonpositive S-curvature, then
Thus from Lemma 2.4, we get the following.
Proposition 3.2 Let be a Finsler n-manifold with nonpositive S-curvature. If the flag curvature satisfies , then the following holds whenever the distance function ρ is smooth:
4 Volume comparison theorems
Let be a Finsler n-manifold. For a fixed point , define
Then . Let be the local coordinates that are intrinsic to . For any , the polar coordinates of q are defined by , where , and . Since , we conclude
in view of the Gauss lemma. Therefore, if , then from the definition of the Finsler-Laplacian of a function we have
Proposition 4.1 Let be a complete Finsler n-manifold with nonpositive S-curvature. If the flag curvature satisfies , then the function
is monotone increasing for , where is the injectivity radius of p and denotes the geodesic ball of radius ρ in a space form of constant sectional curvature c. In particular, for the Busemann-Hausdorff volume form , one has
Proof By (4.1), Proposition 3.2 and the assumption of Proposition 4.1, we have
which implies that the function
is monotone increasing with respect to ρ, where
Let . It is easy to see that for . Set
Then for ,
For two positive integrable functions f and g, if is monotone increasing, then the function
is also monotone increasing (see Lemma 5.1 in  for details). From this statement, one finds that is monotone increasing, and also the function
is monotone increasing for .
To prove (4.2), we only need to show
when . Since
it is sufficient to prove
which can be directly obtained from . □
By using Theorem 3.1, we can get the following result similarly.
Proposition 4.2 Let be a complete and simply connected Finsler n-manifold with nonpositive flag curvature and nonpositive S-curvature. If the weighted Ricci curvature satisfies , then the function
is monotone increasing. In particular, for the Busemann-Hausdorff volume form , one has
Define reversibility as follows:
Obviously, , and if and only if is reversible.
In what follows, we shall generalize Calabi-Yau’s linear volume growth theorem to Finsler manifolds with an arbitrary volume form.
Theorem 4.3 Let be a complete noncompact Finsler n-manifold with finite reversibility λ. If the weighted Ricci curvature satisfies , , then
where (resp. ) denotes the forward (resp. backward) geodesic ball of radius R centered at p and C denotes the constant depending on N, λ, and (resp. ).
Proof Let be a given point. Namely, . Let ρ be the distance function . Then . From Lemma 2.3 we have
Therefore, for any nonnegative function , one obtains
for any . If , then is a Lipschitz continuous function and . Since the Stokes formula still holds for Lipschitz continuous functions, we have
It follows from (4.4) that
Notice that , , it is easy to find from the triangle inequality that
Therefore, from (4.7) and (4.8) we have
On the other hand, it is not hard to see that . Combining this and (4.9) one obtains
Replacing by R, we have
Next, we consider the second part of Theorem 4.3. Let be the reverse Finsler metric of F. If F reversible, then . We put an arrow ← on those quantities associated with . For example,
If the weighted Ricci curvature of F satisfies , then for the reverse Finsler metric , . Moreover, the corresponding Laplacian comparison theorem still holds . Since the curvature condition is common between F and , the assertion for w.r.t. F follows from that for w.r.t. . So the proof is omitted here. □
Corollary 4.4 A complete noncompact Finsler n-manifold with nonnegative weighted Ricci curvature and finite reversibility must have infinite volume.
Theorem 4.5 Let be a complete and simply connected Finsler n-manifold, with nonpositive flag curvature and nonpositive S-curvature. If the weighted Ricci curvature satisfies (), then for any fixed , there exists a positive constant such that when , one has
where is the volume of the forward geodesic ball centered at with radius ρ.
Proof Using (3.1), Lemma 2.2, and Definition 2.1, we have
Combining (4.1) one gets
which together with (3.2) and (3.4) yields
On the other hand, for a Riemannian manifold with constant sectional curvature , we have and
Set . Then
By (4.11) and (4.13), and the assumption of Theorem 4.5, we have
Therefore is increasing in ρ. Hence when ,
Notice that , (). We obtain , which means that is also increasing in ρ. It is well known that
Thus when , one has
as , which shows that there exists such that
where denotes the volume of the unit sphere . □
By similar argument, we also have the following result.
Proposition 4.6 Let be a complete and simply connected Finsler n-manifold with nonpositive S-curvature. If the flag curvature satisfies , then the volume of the forward geodesic ball of M grows at least exponentially.
Remark 4.7 Theorem 4.5 and Proposition 4.6 can also be deduced from Proposition 4.1 and Proposition 4.2.
In , Milnor proved that the fundamental group of a compact Riemannian manifold of negative sectional curvature has exponential growth. Then this result was generalized to the case of negative Ricci curvature and nonpositive sectional curvature in  and . The key point of the proof is to give a lower bound estimate for the volume of the geodesic balls of the universal covering space. In  and , the results were also generalized to the Finsler setting. By using the same method, we get another version of Milnor’s results in Finsler geometry.
Theorem 4.8 Let be a compact Finsler n-manifold with nonpositive S-curvature. Suppose that one of the following two conditions holds:
the flag curvature satisfies ;
M has nonpositive flag curvature and .
Then the fundamental group of M grows at least exponentially.
5 Mckean type eigenvalue estimates
Let be a Finsler manifold, be a domain with compact closure and nonempty boundary ∂ Ω. The first eigenvalue is defined by
where is the completion of . If are bounded domains, then . Thus, if are bounded domains such that , then the limit
exists, and it is independent of the choice of .
Theorem 5.1 Let be a complete noncompact and simply connected Finsler n-manifold with finite reversibility λ and nonpositive S-curvature. If the flag curvature satisfies (), then
Proof For , set , where denotes the forward geodesic ball of radius R centered at p. Then is differentiable in and ∇ρ is a smooth vector field in . Let . Notice that , and we have
where is the dual Finsler metric of F. Since , it is known from Proposition 3.2 that . Hence
holds for any . Integrating both sides of (5.1) over and using the divergence theorem, we obtain
Choosing , one has
Letting , we get
Since f is arbitrary, the formula above means
Note that is a complete noncompact and simply connected Finsler manifold. Letting , we have
By using Theorem 3.1 and a similar argument, we can also prove the following result.
Theorem 5.2 Let be a complete noncompact and simply connected Finsler n-manifold with finite reversibility λ, nonpositive flag curvature and nonpositive S-curvature. If the weighted Ricci curvature satisfies (), then
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This work is supported partly by NNSFC (no. 10971239, 11171253) and NSFHE (KJ2012B197). The authors would like to thank the referees for their useful comments on the manuscript.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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Yin, S., He, Q. & Zheng, D. Some comparison theorems and their applications in Finsler geometry. J Inequal Appl 2014, 107 (2014). https://doi.org/10.1186/1029-242X-2014-107
- comparison theorem
- weighted Ricci curvature