Hindawi Advances in Mathematical Physics Volume 2018, Article ID 1761608, 4 pages https://doi.org/10.1155/2018/1761608
Research Article A Note on Finsler Version of Calabi-Yau Theorem Songting Yin ,1,2 Ruixin Wang,3 and Pan Zhang
3,4
1
Department of Mathematics and Computer Science, Tongling University, Tongling, Anhui 244000, China 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 2
Correspondence should be addressed to Pan Zhang;
[email protected] Received 18 April 2018; Accepted 13 August 2018; Published 2 September 2018 Academic Editor: Antonio Scarfone Copyright Β© 2018 Songting Yin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We generalize Calabi-Yauβs 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 (π, πΉ, ππ) is a differential manifold equipped with a Finsler metric πΉ and a volume form ππ. 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 [1β5], 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 β₯ βπΆ/π(π₯)2 for some constant πΆ, where π(π₯) is the distance function from some fixed point π (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 (π, πΉ, ππ) be a complete noncompact Finsler n-manifold with finite reversibility π. Assume that π(π₯) = ππΉ(π, π₯) is the distance function from a fixed point π β π. If the weighted Ricci curvature satisfies Ricπ(π₯, βπ) β₯ βπΆπβ2 (π₯) for some real number π β [π, +β) and some positive constant C, then
ππ Μ volπΉ (π΅+π (π
)) β₯ πΆπ
, ππ
Μ volπΉ (π΅βπ (π
)) β₯ πΆπ
,
(1)
where π΅+π (π
)(resp., π΅βπ (π
)) denotes the forward (resp., backΜ is some conward) geodesic ball of radius R centered at p and πΆ ππ + ππ stant depending on π, πΆ, π, volπΉ (π΅π (1)) (resp., volπΉ (π΅βπ (1))). Thus, the manifold must have infinite volume. Remark 2. Theorem 1 does not coincide with that of the weighted Riemannian manifold (π, πβπ ) since the weighted Ricci curvature Ricπ(π₯, π¦) and Finsler geodesic balls do not coincide with those Ricβπ π (π₯, βπ) 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 π(π₯) = ππΉ(π, π₯) be the forward distance function from π β π and consider the weighted Riemannian metric πβπ (smooth on π\(πΆπ’π‘(π) βͺ π)). Then we apply the Riemannian calculation for πβπ (in π\(πΆπ’π‘(π)βͺπ) to be precise) and obtain a nonlinear FinslerLaplacian comparison result under certain condition. Next we construct a trial function π and use it to estimate β«π πΞπ2 .
2
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Finally using containing relation of the geodesic balls, we can prove Theorem 1, as required. Let (π, πΉ, ππ) be a Finsler π-manifold. For π β ππ₯ π\0, define π (π₯, π) fl log
βdet (πππ (π₯, π)) π (π₯)
.
π [π (πΎΜ (π‘))]π‘=0 , ππ‘
π [π (πΎ (π‘) , πΎΜ (π‘))]π‘=0 . ππ‘
π=1
(8)
β€ βπΌπΎ (ππ , ππ ) .
(3)
Thus, direct computation gives Ξπ|π = trπβπ π» (π)|π β π (π, βπ (π)) β€ β
β«
Then the weighted Ricci curvature of (π, πΉ, ππ) is defined by (see [11])
+
{Ric (π) + π Μ (π) , for π (π) = 0, Ricπ (π) fl { ββ, otherwise, { S (π)2 , (π β π) πΉ (π)2
π(π)
0
=
=
ππ‘ π(π)
π(π)
0
We first give an upper estimate for the Laplacian of the distance function. Theorem 3. Let (π, πΉ, ππ) be a Finsler n-manifold. Assume that π(π₯) = ππΉ(π, π₯) is the forward distance function from a fixed point π β π. If the weighted Ricci curvature satisfies Ricπ(π₯, βπ) β₯ βπΆπβ2 (π₯) for some real number π β [π, +β) and some positive constant C, then
{π‘2 (Ricπ (π (π‘)) +
β
β«
π(π)
0
β
β«
π(π)
π‘2 Ricπ (π (π‘)) ππ‘ β
Proof. Suppose that π(π₯) is smooth at π β π. Let πΎ : [0, π(π)] σ³¨β π be a regular minimal geodesic from π to π and denote its tangent vector by π = πΎ.Μ Choose a ππ-orthonormal basis {π1 , . . . , ππβ1 , ππ = π} at π. Then, by paralleling them along πΎ, we obtain π parallel vector fields {πΈ1 (π‘), . . . , πΈπβ1 (π‘), πΈπ (π‘) = π}. For any π β {1, . . . , π}, one can get a unique Jacobi vector field along πΎ satisfying π½π (0) = 0, π½π (π(π)) = ππ . Set ππ (π‘) = (π‘/π(π))πΈπ (π‘). Then ππ (0) = π½π (0) = 0, ππ (π(π)) = π½π (π(π)). Recall that the Hessian of π is βπ, π β ππ₯ π.
(7)
π(π)
0
(9)
1
2
π (π)
2
{(
π‘π (πΎ (π‘) , π (π‘)) + βπ β π) βπ β π
β (π β π)} ππ‘ β€ β
β«
2
π (πΎ (π‘) , π (π‘)) ) πβπ
πβ1 1 β π (π) π (π)2
(6)
pointwise on π\({π}βͺπΆπ’π‘(π)) and in the sense of distributions on π\{π}.
πβ1 1 β π (π) π (π)2
{π‘2 (Ric (π (π‘)) + π Μ (πΎ (π‘) , π (π‘))) + 2π‘π} ππ‘
+ 2π‘π} ππ‘ =
0
πβ1+πΆ Ξπ β€ π
} ππ‘ =
πβ1 1 β π (π) π (π)2
β
β«
Ricβ (π) fl Ric (π) + πΜ (π) .
2
{(π β 1) β Ric (π (π‘)) π‘2 } ππ‘ β π (π, βπ (π))
π (π‘2 π (πΎ (π‘) , π (π‘)))
0
(5)
1 π (π)
π(π) 1 πβ1 β β« {Ric (π (π‘)) π‘2 2 π (π) π (π) 0
β
β«
βπ β (π, β) ,
π» (π) (π, π) = ππ (π) β π·βπ π π (π) ,
π=1 π
(4)
Ricπ (π) fl Ric (π) + πΜ (π) β
π
π=1
where π β ππ₯ π, and πΎ : (βπ, π) σ³¨β π is the geodesic with Μ = π. π is called the π-curvature [2]. Following πΎ(0) = π₯, πΎ(0) [11], we define π Μ (π) fl πΉβ2 (π)
π
σ΅¨ trπβπ π» (π)σ΅¨σ΅¨σ΅¨σ΅¨π = βπ» (π) (ππ , ππ ) = βπΌπΎ (π½π , π½π )
(2)
π is called the distortion of (π, πΉ, ππ). To measure the rate of distortion along geodesics, we define π (π₯, π) fl
Then, by basic index lemma, we obtain (see [3])
1 πβ1 β 2 π (π) π (π)
π‘2 Ricπ (π (π‘)) ππ‘,
where in the fourth expression we use the fact πΉ(π(π‘)) = 1. Note that Ricπ(π₯, βπ) β₯ βπΆπβ2 (π₯). This together with (9) yields Ξπ|π β€
πβ1+πΆ . π (π)
(10)
Now by a standard way, it is not difficult to verify that the inequality above holds in the distributional sense on π\{π}.
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3
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 πβ1+πΆ (11) , Ξπ β€ π which yields Ξβπ π2 = 2πΞπ + 2πΉ (βπ)2 β€ 2 (π β 1 + πΆ) + 2 = 2 (π + πΆ) .
Notice that ππΉ(π, π) β€ πππΉ (π, π), βπ, π β π. From the triangle inequality, one has π΅+π (1) β π΅+π₯0 (π
+ 1) \π΅+π₯0 (π
β π) ,
π
ππ
2 (π + πΆ) volπΉ (π΅+π₯0 (π
+ 1)) (12)
(13)
π
Let π₯0 β ππ΅βπ (π
) be a given point. Then, ππΉ(π₯0 , π) = π
. Set 1, 0 β€ π‘ β€ π
β π; { { { {π
+ 1 β π‘ π (π‘) fl { , π
β π β€ π‘ β€ π
+ 1; { { 1+π { π‘ β₯ π
+ 1, {0,
πΉ (π)
πβππ\0 πΉ (βπ)
.
β₯
2 (π
β π) ππ + volπΉ (π΅π₯0 (π
+ 1) \π΅+π₯0 (π
β π)) 1+π
β₯
2 (π
β π) ππ + volπΉ π΅π (1) . 1+π
On the other hand, it is not hard to see π΅+π₯0 (π
+ 1) β π΅+π ((π + 1)(π
+ 1)). Combining this and the formula (20) yields volπΉ (π΅+π ((π + 1) (π
+ 1)))
(14)
β₯
(15)
βπ, π β π.
βπ
β« πΞ π ππ = β β«
π΅+π₯0 (π
+1)
π
= β2 β«
π΅+π₯0 (π
+1)
ππ
volπΉ (π΅+π (π
)) β₯
(17)
=
2 β« πππ 1 + π π΅+π₯0 (π
+1)\π΅+π₯0 (π
βπ)
β₯
2 (π
β π) ππ + volπΉ (π΅π₯0 (π
+ 1) \π΅+π₯0 (π
β π)) , 1+π
β€ 2 (π + πΆ) β« πππ = 2 (π + πΆ) β«
π΅+π₯0 (π
+1)
ππ
= 2 (π + πΆ) volπΉ (π΅+π₯0 (π
+ 1)) .
(22)
(1))) π
.
Data Availability No data were used to support this study.
This paper is based on 2010 Mathematics Subject Classification (Primary 53C60; Secondary 53C24).
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).
2 (π
β π) ππ + volπΉ (π΅π₯0 (π
+ 1) \π΅+π₯0 (π
β π)) 1+π
ππ
(π΅+π
Conflicts of Interest
which together with (13) gives
π΅+π₯0 (π
+1)
Μ (π, πΆ, π, volππ πΆ πΉ
Additional Points
πσΈ (π (π₯)) ππΉ (βπ)2 ππ
β€ 2 (π + πΆ) β«
β₯
(π
/ (π + 1) β (1 + π)) ππ + volπΉ π΅π (1) (π + πΆ) (1 + π)
βπ 2
πβπ (β π, β π ) ππ
π
(21)
(16)
If π(π₯) = π(π(π₯)), then π(π₯) is a Lipschitz continuous function and supp π β π΅+π₯0 (π
+ 1). Since Stokes formula still holds for Lipschitz continuous functions, we have βπ 2
(π
β π) ππ volπΉ π΅+π (1) . (π + πΆ) (1 + π)
Replacing (π + 1)(π
+ 1) by π
, we have
(π, πΉ) is called reversible if π = 1. It is clear that the distance function ππΉ of πΉ satisfies ππΉ (π, π) β€ πππΉ (π, π) ,
(20)
ππ
for any π
> π, where π is the reversibility of πΉ defined by (see [12]) π = max
(19)
Therefore, from (18) and (19) we have
Therefore, for any nonnegative function π β πΆβ 0 (π), it holds that β« πΞβπ π2 ππ β€ 2 (π + πΆ) β« πππ.
βπ
> π.
References
πππ (18)
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