A Note on Finsler Version of Calabi-Yau Theorem

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.


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][2][3][4][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 Ric ≥ −/() 2 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.

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 ∫  Δ 2 .

Advances in Mathematical Physics
Finally using containing relation of the geodesic balls, we can prove Theorem 1, as required.
Note that Ric  (, ∇) ≥ − −2 ().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 yields Therefore, for any nonnegative function  ∈  ∞ 0 (), it holds that Let  0 ∈  −  () be a given point.Then,   ( 0 , ) = .Set for any  > , where  is the reversibility of  defined by (see [12])