# A Note on Finsler Version of Calabi-Yau Theorem

Authors: Songting Yin; Ruixin Wang; Pan Zhang
Journal: Advances in Mathematical Physics
 (2018)

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## 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.


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## Body

## 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.
Theorem1 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 Theorems1, 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 thedistortion 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 theweighted 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
rpointwise on M
\
(
{
p
}
∪
C
u
t
(
p
)
) and in the sense of distributions on M
\
{
p
}.Proof.
Suppose thatr
(
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 Theorem1.
We only prove the first inequality as the second one can be proved in a similar way. From Theorem3 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
.



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*Source: 1761608-2018-09-02.xml*