A Note on Finsler Version of Calabi-Yau Theorem

Abstract

We generalize Calabi-Yau’s linear volume growth theorem to Finsler manifold with the weighted Ricci curvature bounded below by a negative function and show that such a manifold must have infinite volume.

1. Introduction

A Finsler space is a differential manifold equipped with a Finsler metric and a volume form . The class of Finsler spaces is one of the most important metric measure spaces. Up to now, Finsler geometry has developed rapidly in its global and analytic aspects. In [1–5], the study was well implemented on Laplacian comparison theorem, Bishop-Gromov volume comparison theorem, Liouville-type theorem, and so on.

A theorem due to Calabi and Yau states that the volume of any complete noncompact Riemannian manifold with nonnegative Ricci curvature has at least linear growth (see [6, 7]). The result was generalized to Riemannian manifolds with lower bound for some constant , where is the distance function from some fixed point (see [8, 9]). As to the Finsler case, if the (weighted) Ricci curvature is nonnegative, the Calabi-Yau type linear volume growth theorem was obtained in [4, 10]. Therefore, it is natural to generalize it in the Finsler setting with the weighted Ricci curvature bounded below by a negative function. Our main result is as follows.

Theorem 1. Let be a complete noncompact Finsler n-manifold with finite reversibility . Assume that is the distance function from a fixed point . If the weighted Ricci curvature satisfies for some real number and some positive constant C, thenwhere (resp., ) denotes the forward (resp., backward) geodesic ball of radius R centered at p and is some constant depending on (resp., ). Thus, the manifold must have infinite volume.

Remark 2. Theorem 1 does not coincide with that of the weighted Riemannian manifold since the weighted Ricci curvature and Finsler geodesic balls do not coincide with those and Riemannian geodesic balls in weighted Riemannian manifold.

2. The Proof of the Main Theorem

To prove Theorems 1, we need to obtain a Laplacian comparison theorem on the Finsler manifold and then follow the method of Schoen and Yau in [7] (see also [9]). We have to adapt the arguments and give some adjustments in the Finsler setting. Specifically, let be the forward distance function from and consider the weighted Riemannian metric (smooth on ). Then we apply the Riemannian calculation for (in to be precise) and obtain a nonlinear Finsler-Laplacian comparison result under certain condition. Next we construct a trial function and use it to estimate . Finally using containing relation of the geodesic balls, we can prove Theorem 1, as required.

Let be a Finsler -manifold. For , define is called the distortion of . To measure the rate of distortion along geodesics, we define where , and is the geodesic with . is called the -curvature [2]. Following [11], we define

Then the weighted Ricci curvature of is defined by (see [11])

We first give an upper estimate for the Laplacian of the distance function.

Theorem 3. Let be a Finsler n-manifold. Assume that is the forward distance function from a fixed point . If the weighted Ricci curvature satisfies for some real number and some positive constant C, then pointwise on and in the sense of distributions on .

Proof. Suppose that is smooth at . Let be a regular minimal geodesic from to and denote its tangent vector by . Choose a -orthonormal basis at . Then, by paralleling them along , we obtain parallel vector fields . For any , one can get a unique Jacobi vector field along satisfying . Set . Then . Recall that the Hessian of is Then, by basic index lemma, we obtain (see [3]) Thus, direct computation giveswhere in the fourth expression we use the fact . Note that . This together with (9) yields Now by a standard way, it is not difficult to verify that the inequality above holds in the distributional sense on .

Proof of Theorem 1. We only prove the first inequality as the second one can be proved in a similar way. From Theorem 3 one obtains which yieldsTherefore, for any nonnegative function , it holds thatLet be a given point. Then, . Setfor any , where is the reversibility of defined by (see [12]) is called reversible if . It is clear that the distance function of satisfies If , then is a Lipschitz continuous function and . Since Stokes formula still holds for Lipschitz continuous functions, we havewhich together with (13) givesNotice that . From the triangle inequality, one hasTherefore, from (18) and (19) we haveOn the other hand, it is not hard to see . Combining this and the formula (20) yields Replacing by , we have

Data Availability

No data were used to support this study.

Additional Points

This paper is based on 2010 Mathematics Subject Classification (Primary 53C60; Secondary 53C24).

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This project is supported by AHNSF (no. 1608085MA03), KLAMFJPU (no. SX201805), and NNSFC (no. 11471246).