1. Introduction
A Finsler space
(
M
,
F
,
d
μ
)
is a differential manifold equipped with a Finsler metric
F
and a volume form
d
μ
. 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
R
i
c
≥
-
C
/
r
(
x
)
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
μ
)
be a complete noncompact Finsler n-manifold with finite reversibility
η
. Assume that
r
(
x
)
=
d
F
(
p
,
x
)
is the distance function from a fixed point
p
∈
M
. If the weighted Ricci curvature satisfies
Ric
N
(
x
,
∇
r
)
≥
-
C
r
-
2
(
x
)
for some real number
N
∈
[
n
,
+
∞
)
and some positive constant C, then
(1)
vol
F
d
μ
B
p
+
R
≥
C
~
R
,
vol
F
d
μ
B
p
-
R
≥
C
~
R
,
where
B
p
+
(
R
)
(resp.,
B
p
-
(
R
)
) denotes the forward (resp., backward) geodesic ball of radius R centered at p and
C
~
is some constant depending on
N
,
C
,
η
,
vol
F
d
μ
(
B
p
+
(
1
)
)
(resp.,
vol
F
d
μ
(
B
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
∇
r
)
since the weighted Ricci curvature
R
i
c
N
(
x
,
y
)
and Finsler geodesic balls do not coincide with those
R
i
c
N
∇
r
(
x
,
∇
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
F
(
p
,
x
)
be the forward distance function from
p
∈
M
and consider the weighted Riemannian metric
g
∇
r
(smooth on
M
\
(
C
u
t
(
p
)
∪
p
)
). Then we apply the Riemannian calculation for
g
∇
r
(in
M
\
(
C
u
t
(
p
)
∪
p
)
to be precise) and obtain a nonlinear Finsler-Laplacian comparison result under certain condition. Next we construct a trial function
φ
and use it to estimate
∫
M
φ
Δ
r
2
. Finally using containing relation of the geodesic balls, we can prove Theorem 1, as required.
Let
(
M
,
F
,
d
μ
)
be a Finsler
n
-manifold. For
V
∈
T
x
M
\
0
, define
(2)
τ
x
,
V
≔
log
det
g
i
j
x
,
V
σ
x
.
τ
is called the distortion of
(
M
,
F
,
d
μ
)
. To measure the rate of distortion along geodesics, we define
(3)
S
x
,
V
≔
d
d
t
τ
γ
˙
t
t
=
0
,
where
V
∈
T
x
M
, and
γ
:
(
-
ε
,
ε
)
→
M
is the geodesic with
γ
(
0
)
=
x
,
γ
˙
(
0
)
=
V
.
S
is called the
S
-curvature [2]. Following [11], we define
(4)
S
˙
V
≔
F
-
2
V
d
d
t
S
γ
t
,
γ
˙
t
t
=
0
.
Then the weighted Ricci curvature of
(
M
,
F
,
d
μ
)
is defined by (see [11])
(5)
R
i
c
n
V
≔
R
i
c
V
+
S
˙
V
,
for
S
V
=
0
,
-
∞
,
otherwise,
R
i
c
N
V
≔
R
i
c
V
+
S
˙
V
-
S
V
2
N
-
n
F
V
2
,
∀
N
∈
n
,
∞
,
R
i
c
∞
V
≔
R
i
c
V
+
S
˙
V
.
We first give an upper estimate for the Laplacian of the distance function.
Theorem 3.
Let
(
M
,
F
,
d
μ
)
be a Finsler n-manifold. Assume that
r
(
x
)
=
d
F
(
p
,
x
)
is the forward distance function from a fixed point
p
∈
M
. If the weighted Ricci curvature satisfies
Ric
N
(
x
,
∇
r
)
≥
-
C
r
-
2
(
x
)
for some real number
N
∈
[
n
,
+
∞
)
and some positive constant C, then
(6)
Δ
r
≤
N
-
1
+
C
r
pointwise on
M
\
(
{
p
}
∪
C
u
t
(
p
)
)
and in the sense of distributions on
M
\
{
p
}
.
Proof.
Suppose that
r
(
x
)
is smooth at
q
∈
M
. Let
γ
:
[
0
,
r
(
q
)
]
→
M
be a regular minimal geodesic from
p
to
q
and denote its tangent vector by
T
=
γ
˙
. Choose a
g
T
-orthonormal basis
{
e
1
,
…
,
e
n
-
1
,
e
n
=
T
}
at
q
. Then, by paralleling them along
γ
, we obtain
n
parallel vector fields
{
E
1
(
t
)
,
…
,
E
n
-
1
(
t
)
,
E
n
(
t
)
=
T
}
. For any
i
∈
{
1
,
…
,
n
}
, one can get a unique Jacobi vector field along
γ
satisfying
J
i
(
0
)
=
0
,
J
i
(
r
(
q
)
)
=
e
i
. Set
W
i
(
t
)
=
(
t
/
r
q
)
E
i
(
t
)
. Then
W
i
(
0
)
=
J
i
(
0
)
=
0
,
W
i
(
r
(
q
)
)
=
J
i
(
r
(
q
)
)
. Recall that the Hessian of
r
is
(7)
H
r
X
,
Y
=
X
Y
r
-
D
X
∇
r
Y
r
,
∀
X
,
Y
∈
T
x
M
.
Then, by basic index lemma, we obtain (see [3])
(8)
t
r
g
∇
r
H
r
q
=
∑
i
=
1
n
H
r
e
i
,
e
i
=
∑
i
=
1
n
I
γ
J
i
,
J
i
≤
∑
i
=
1
n
I
γ
W
i
,
W
i
.
Thus, direct computation gives
(9)
Δ
r
q
=
t
r
g
∇
r
H
r
q
-
S
q
,
∇
r
q
≤
1
r
q
2
∫
0
r
q
n
-
1
-
R
i
c
T
t
t
2
d
t
-
S
q
,
∇
r
q
=
n
-
1
r
q
-
1
r
q
2
∫
0
r
q
R
i
c
T
t
t
2
+
d
t
2
S
γ
t
,
T
t
d
t
d
t
=
n
-
1
r
q
-
1
r
q
2
∫
0
r
q
t
2
R
i
c
T
t
+
S
˙
γ
t
,
T
t
+
2
t
S
d
t
=
n
-
1
r
q
-
1
r
q
2
∫
0
r
q
t
2
R
i
c
N
T
t
+
S
γ
t
,
T
t
2
N
-
n
+
2
t
S
d
t
=
n
-
1
r
q
-
1
r
q
2
∫
0
r
q
t
2
R
i
c
N
T
t
d
t
-
1
r
q
2
∫
0
r
q
t
S
γ
t
,
T
t
N
-
n
+
N
-
n
2
-
N
-
n
d
t
≤
N
-
1
r
q
-
1
r
q
2
∫
0
r
q
t
2
R
i
c
N
T
t
d
t
,
where in the fourth expression we use the fact
F
(
T
(
t
)
)
=
1
. Note that
R
i
c
N
(
x
,
∇
r
)
≥
-
C
r
-
2
(
x
)
. This together with (9) yields
(10)
Δ
r
q
≤
N
-
1
+
C
r
q
.
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
(11)
Δ
r
≤
N
-
1
+
C
r
,
which yields
(12)
Δ
∇
r
r
2
=
2
r
Δ
r
+
2
F
∇
r
2
≤
2
N
-
1
+
C
+
2
=
2
N
+
C
.
Therefore, for any nonnegative function
φ
∈
C
0
∞
(
M
)
, it holds that
(13)
∫
M
φ
Δ
∇
r
r
2
d
μ
≤
2
N
+
C
∫
M
φ
d
μ
.
Let
x
0
∈
∂
B
p
-
(
R
)
be a given point. Then,
d
F
(
x
0
,
p
)
=
R
. Set
(14)
ψ
t
≔
1
,
0
≤
t
≤
R
-
η
;
R
+
1
-
t
1
+
η
,
R
-
η
≤
t
≤
R
+
1
;
0
,
t
≥
R
+
1
,
for any
R
>
η
, where
η
is the reversibility of
F
defined by (see [12])
(15)
η
=
max
X
∈
TM
\
0
F
X
F
-
X
.
(
M
,
F
)
is called reversible if
η
=
1
. It is clear that the distance function
d
F
of
F
satisfies
(16)
d
F
p
,
q
≤
η
d
F
q
,
p
,
∀
p
,
q
∈
M
.
If
φ
(
x
)
=
ψ
(
r
(
x
)
)
, then
φ
(
x
)
is a Lipschitz continuous function and
s
u
p
p
φ
⊂
B
x
0
+
(
R
+
1
)
. Since Stokes formula still holds for Lipschitz continuous functions, we have
(17)
∫
M
φ
Δ
∇
r
r
2
d
μ
=
-
∫
B
x
0
+
R
+
1
g
∇
r
∇
∇
r
φ
,
∇
∇
r
r
2
d
μ
=
-
2
∫
B
x
0
+
R
+
1
ψ
′
r
x
r
F
∇
r
2
d
μ
=
2
1
+
η
∫
B
x
0
+
R
+
1
\
B
x
0
+
R
-
η
r
d
μ
≥
2
R
-
η
1
+
η
v
o
l
F
d
μ
B
x
0
+
R
+
1
\
B
x
0
+
R
-
η
,
which together with (13) gives
(18)
2
R
-
η
1
+
η
v
o
l
F
d
μ
B
x
0
+
R
+
1
\
B
x
0
+
R
-
η
≤
2
N
+
C
∫
M
φ
d
μ
=
2
N
+
C
∫
B
x
0
+
R
+
1
φ
d
μ
≤
2
N
+
C
∫
B
x
0
+
R
+
1
d
μ
=
2
N
+
C
v
o
l
F
d
μ
B
x
0
+
R
+
1
.
Notice that
d
F
(
p
,
q
)
≤
η
d
F
(
q
,
p
)
,
∀
p
,
q
∈
M
. From the triangle inequality, one has
(19)
B
p
+
1
⊂
B
x
0
+
R
+
1
\
B
x
0
+
R
-
η
,
∀
R
>
η
.
Therefore, from (18) and (19) we have
(20)
2
N
+
C
v
o
l
F
d
μ
B
x
0
+
R
+
1
≥
2
R
-
η
1
+
η
v
o
l
F
d
μ
B
x
0
+
R
+
1
\
B
x
0
+
R
-
η
≥
2
R
-
η
1
+
η
v
o
l
F
d
μ
B
p
+
1
.
On the other hand, it is not hard to see
B
x
0
+
(
R
+
1
)
⊂
B
p
+
(
(
η
+
1
)
(
R
+
1
)
)
. Combining this and the formula (20) yields
(21)
v
o
l
F
d
μ
B
p
+
η
+
1
R
+
1
≥
R
-
η
N
+
C
1
+
η
v
o
l
F
d
μ
B
p
+
1
.
Replacing
(
η
+
1
)
(
R
+
1
)
by
R
, we have
(22)
v
o
l
F
d
μ
B
p
+
R
≥
R
/
η
+
1
-
1
+
η
N
+
C
1
+
η
v
o
l
F
d
μ
B
p
+
1
≥
C
~
N
,
C
,
η
,
v
o
l
F
d
μ
B
p
+
1
R
.