# A Note on Finsler Version of Calabi-Yau Theorem.

1. Introduction

A Finsler space (M, F, d[mu]) is a differential manifold equipped with a Finsler metric F and a volume form d[mu]. 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 Ric [greater than or equal to] -C/r[(x).sup.2] for some constant C, where r(x) is the distance function from some fixed point p (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 (M, F, d[mu]) be a complete noncompact Finsler n-manifold with finite reversibility [eta]. Assume that r(x) = [d.sub.F](p, x) is the distance function from a fixed point p [member of] M. If the weighted Ricci curvature satisfies [Ric.sub.N](x, [nabla]r) [greater than or equal to] -[Cr.sup.-2](x) for some real number N [member of] [n, +[infinity]) and some positive constant C, then

[vol.sup.d[mu].sub.F] ([B.sup.+.sub.p] (R)) [greater than or equal to] [??]R,

[vol.sup.d[mu].sub.F] ([B.sup.-.sub.p] (R)) [greater than or equal to] [??]R, (1)

where [B.sup.+.sub.p](R)(resp., [B.sup.-.sub.p] (R)) denotes the forward (resp., back ward) geodesic ball of radius R centered at p and [??] is some constant depending on N, C, [eta], [vol.sup.d[mu].sub.F] ([B.sup.+.sub.p](1)) (resp., [vol.sup.d[mu].sub.F] ([B.sup.- .sub.p](1))). Thus, the manifold must have infinite volume.

Remark 2. Theorem 1 does not coincide with that of the weighted Riemannian manifold (M, [g.sub.[nabla]r]) since the weighted Ricci curvature [Ric.sub.N](x, y) and Finsler geodesic balls do not coincide with those [Ric.sup.[nabla]r.sub.N][(x, [nabla]r) 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 r(x) = [d.sub.F](p, x) be the forward distance function from p [member of] M and consider the weighted Riemannian metric [g.sub.[nabla]r] (smooth on M\(Cut(p) [union] p)). Then we apply the Riemannian calculation for gVr (in M\(Cut(p) [union] p) to be precise) and obtain a nonlinear Finsler-Laplacian comparison result under certain condition. Next we construct a trial function [phi] and use it to estimate [[integral].sub.M] [phi][DELTA][r.sup.2]. Finally using containing relation of the geodesic balls, we can prove Theorem 1, as required.

Let (M, F, d[mu]) be a Finsler n-manifold. For V [member of] [T.sub.x]M\0, define

[tau] (X, V) := log [square root of (det ([g.sub.ij] (x, V)))]/[sigma] (x). (2)

[tau] is called the distortion of (M, F, d[mu]). To measure the rate of distortion along geodesics, we define

S(x, V) := d/dt [[[tau]([??] (t))].sub.t=0], (3)

where V [member of] [T.sub.x]M, and [gamma] : (-[epsilon], [epsilon]) [right arrow] M is the geodesic with [gamma](0) = x, [??](0) = V. S is called the S-curvature [2]. Following [11], we define

[mathematical expression not reproducible] (4)

Then the weighted Ricci curvature of (M, F, d[mu]) is defined by (see [11])

[mathematical expression not reproducible] (5)

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

Theorem 3. Let (M, F, d[mu]) be a Finsler n-manifold. Assume that r(x) = [d.sub.F](p, x) is the forward distance function from a fixed point p [member of] M. If the weighted Ricci curvature satisfies [Ric.sub.N](x, [nabla]r) [greater than or equal to] -[Cr.sup.-2] (x) for some real number N [member of] [n, +[infinity]) and some positive constant C, then

[DELTA]r [less than or equal to] N - 1 + C/r (6)

pointwise on M\({p} [union] Cut(p)) and in the sense of distributions on M\[p}.

Proof. Suppose that r(x) is smooth at q [member of] M. Let [gamma] : [0, r(q)] [right arrow] M be a regular minimal geodesic from p to q and denote its tangent vector by T = [??]. Choose a [g.sub.T]-orthonormal basis {[e.sub.1], ..., [e.sub.n-1], [e.sub.n] = T} at q. Then, by paralleling them along [gamma], we obtain n parallel vector fields {[E.sub.1](t), ..., [E.sub.n-1](t), [E.sub.n](t) = T}. For any i [member of] {1, ..., n}, one can get a unique Jacobi vector field along [gamma] satisfying [J.sub.i](0) = 0, [J.sub.i](r(q)) = [e.sub.i]. Set [W.sub.i](t) = (t/r(q))[E.sub.i](t). Then [W.sub.i](0) = [J.sub.i](0) = 0, [W.sub.i](r(q)) = [J.sub.i](r(q)). Recall that the Hessian of r is

H (r) (X, Y) = XY (r) - [D.sup.[nabla]r.sub.X]Y (r), [for all]X, Y [member of] [T.sub.x]M. (7)

Then, by basic index lemma, we obtain (see [3])

[mathematical expression not reproducible] (8)

Thus, direct computation gives

[mathematical expression not reproducible] (9)

where in the fourth expression we use the fact F(T(t)) = 1. Note that [Ric.sub.N](x, [nabla]r) [greater than or equal to] -[Cr.sup.-2](x). This together with (9) yields

[DELTA]r[|.sub.q] [less than or equal to] N - 1 + C/r(q). (10)

Now by a standard way, it is not difficult to verify that the inequality above holds in the distributional sense on M\{p}.

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

[DELTA]r [less than or equal to] N - 1 + C/r, (11)

which yields

[[DELTA].sup.[nabla]r][r.sup.2] = 2r[DELTA]r + 2F [([nabla]r).sup.2] [less than or equal to] 2(N - 1 + C) + 2 = 2(N + C). (12)

Therefore, for any nonnegative function [phi] [member of] [C.sup.[infinity].sub.0](M), it holds that

[[integral].sub.M] [phi][[DELTA].sup.[nabla]r] [r.sup.2] d[mu] [less than or equal to] 2 (N + C) [[integral].sub.M] [phi]d[mu]. (13)

Let [x.sub.0] [member of] [partial derivative][B.sup.-.sub.p](R) be a given point. Then, [d.sub.F]([x.sub.0], p) = R. Set

[mathematical expression not reproducible] (14)

for any R > [eta], where [eta] is the reversibility of F defined by (see [12])

[mathematical expression not reproducible] (15)

(M, F) is called reversible if [eta] = 1. It is clear that the distance function [d.sub.F] of F satisfies

[d.sub.F] (p, q) [less than or equal to] [eta][d.sub.F] (q, p), [for all]p, q [member of] M. (16)

If [phi](x) = [psi](r(x)), then [phi](x) is a Lipschitz continuous function and supp [mathematical expression not reproducible]. Since Stokes formula still holds for Lipschitz continuous functions, we have

[mathematical expression not reproducible] (17)

which together with (13) gives

[mathematical expression not reproducible] (18)

Notice that [d.sub.F](p, q) [less than or equal to] [eta][d.sub.F](q, p), [for all]p, q [member of] M. From the triangle inequality, one has

[mathematical expression not reproducible] (19)

Therefore, from (18) and (19) we have

[mathematical expression not reproducible] (20)

On the other hand, it is not hard to see [mathematical expression not reproducible]. Combining this and the formula (20) yields

[mathematical expression not reproducible] (21)

Replacing ([eta] + 1)(R+ 1) by R, we have

[mathematical expression not reproducible] (22)

https://doi.org/10.1155/2018/1761608

Data Availability

No data were used to support this study.

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).

References

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[9] L. Ma, "Some properties of non-compact complete Riemannian manifolds," Bulletin des Sciences Mathematiques, vol. 130, no. 4, pp. 330-336, 2006.

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Songting Yin (iD), (1,2) Ruixin Wang, (3) and Pan Zhang (iD) (3,4)

(1) Department of Mathematics and Computer Science, Tongling University, Tongling, Anhui 244000, China

(2) Key Laboratory of Applied Mathematics, Putian University, Fujian Province University, Fujian, Putian 351100, China

(3) School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China

(4) School of Mathematics, Sun Yat-sen University, China

Correspondence should be addressed to Pan Zhang; panzhang@mail.ustc.edu.cn

Received 18 April 2018; Accepted 13 August 2018; Published 2 September 2018