A Note on Finsler Version of Calabi-Yau Theorem

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13 Aug 2018 - 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 andΒ ...
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-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.

1. Introduction A Finsler space (𝑀, 𝐹, π‘‘πœ‡) is a differential manifold equipped with a Finsler metric 𝐹 and a volume form π‘‘πœ‡. 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 β‰₯ βˆ’πΆ/π‘Ÿ(π‘₯)2 for some constant 𝐢, where π‘Ÿ(π‘₯) 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, π‘‘πœ‡) be a complete noncompact Finsler n-manifold with finite reversibility πœ‚. Assume that π‘Ÿ(π‘₯) = 𝑑𝐹(𝑝, π‘₯) is the distance function from a fixed point 𝑝 ∈ 𝑀. If the weighted Ricci curvature satisfies Ric𝑁(π‘₯, βˆ‡π‘Ÿ) β‰₯ βˆ’πΆπ‘Ÿβˆ’2 (π‘₯) for some real number 𝑁 ∈ [𝑛, +∞) and some positive constant C, then

π‘‘πœ‡ Μƒ vol𝐹 (𝐡+𝑝 (𝑅)) β‰₯ 𝐢𝑅, π‘‘πœ‡

Μƒ vol𝐹 (π΅βˆ’π‘ (𝑅)) β‰₯ 𝐢𝑅,

(1)

where 𝐡+𝑝 (𝑅)(resp., π΅βˆ’π‘ (𝑅)) denotes the forward (resp., backΜƒ is some conward) geodesic ball of radius R centered at p and 𝐢 π‘‘πœ‡ + π‘‘πœ‡ stant depending on 𝑁, 𝐢, πœ‚, vol𝐹 (𝐡𝑝 (1)) (resp., vol𝐹 (π΅βˆ’π‘ (1))). Thus, the manifold must have infinite volume. Remark 2. Theorem 1 does not coincide with that of the weighted Riemannian manifold (𝑀, π‘”βˆ‡π‘Ÿ ) since the weighted Ricci curvature Ric𝑁(π‘₯, 𝑦) and Finsler geodesic balls do not coincide with those Ricβˆ‡π‘Ÿ 𝑁 (π‘₯, βˆ‡π‘Ÿ) 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‘Ÿ(π‘₯) = 𝑑𝐹(𝑝, π‘₯) be the forward distance function from 𝑝 ∈ 𝑀 and consider the weighted Riemannian metric π‘”βˆ‡π‘Ÿ (smooth on 𝑀\(𝐢𝑒𝑑(𝑝) βˆͺ 𝑝)). Then we apply the Riemannian calculation for π‘”βˆ‡π‘Ÿ (in 𝑀\(𝐢𝑒𝑑(𝑝)βˆͺ𝑝) to be precise) and obtain a nonlinear FinslerLaplacian comparison result under certain condition. Next we construct a trial function πœ‘ and use it to estimate βˆ«π‘€ πœ‘Ξ”π‘Ÿ2 .

2

Advances in Mathematical Physics

Finally using containing relation of the geodesic balls, we can prove Theorem 1, as required. Let (𝑀, 𝐹, π‘‘πœ‡) be a Finsler 𝑛-manifold. For 𝑉 ∈ 𝑇π‘₯ 𝑀\0, define 𝜏 (π‘₯, 𝑉) fl log

√det (𝑔𝑖𝑗 (π‘₯, 𝑉)) 𝜎 (π‘₯)

.

𝑑 [𝜏 (𝛾̇ (𝑑))]𝑑=0 , 𝑑𝑑

𝑑 [𝑆 (𝛾 (𝑑) , 𝛾̇ (𝑑))]𝑑=0 . 𝑑𝑑

𝑖=1

(8)

≀ βˆ‘πΌπ›Ύ (π‘Šπ‘– , π‘Šπ‘– ) .

(3)

Thus, direct computation gives Ξ”π‘Ÿ|π‘ž = trπ‘”βˆ‡π‘Ÿ 𝐻 (π‘Ÿ)|π‘ž βˆ’ 𝑆 (π‘ž, βˆ‡π‘Ÿ (π‘ž)) ≀ β‹…βˆ«

Then the weighted Ricci curvature of (𝑀, 𝐹, π‘‘πœ‡) is defined by (see [11])

+

{Ric (𝑉) + 𝑆 Μ‡ (𝑉) , for 𝑆 (𝑉) = 0, Ric𝑛 (𝑉) fl { βˆ’βˆž, otherwise, { S (𝑉)2 , (𝑁 βˆ’ 𝑛) 𝐹 (𝑉)2

π‘Ÿ(π‘ž)

0

=

=

𝑑𝑑 π‘Ÿ(π‘ž)

π‘Ÿ(π‘ž)

0

We first give an upper estimate for the Laplacian of the distance function. Theorem 3. Let (𝑀, 𝐹, π‘‘πœ‡) be a Finsler n-manifold. Assume that π‘Ÿ(π‘₯) = 𝑑𝐹(𝑝, π‘₯) is the forward distance function from a fixed point 𝑝 ∈ 𝑀. If the weighted Ricci curvature satisfies Ric𝑁(π‘₯, βˆ‡π‘Ÿ) β‰₯ βˆ’πΆπ‘Ÿβˆ’2 (π‘₯) for some real number 𝑁 ∈ [𝑛, +∞) and some positive constant C, then

{𝑑2 (Ric𝑁 (𝑇 (𝑑)) +

β‹…βˆ«

π‘Ÿ(π‘ž)

0

β‹…βˆ«

π‘Ÿ(π‘ž)

𝑑2 Ric𝑁 (𝑇 (𝑑)) 𝑑𝑑 βˆ’

Proof. Suppose that π‘Ÿ(π‘₯) is smooth at π‘ž ∈ 𝑀. Let 𝛾 : [0, π‘Ÿ(π‘ž)] 󳨀→ 𝑀 be a regular minimal geodesic from 𝑝 to π‘ž and denote its tangent vector by 𝑇 = 𝛾.Μ‡ Choose a 𝑔𝑇-orthonormal basis {𝑒1 , . . . , π‘’π‘›βˆ’1 , 𝑒𝑛 = 𝑇} at π‘ž. Then, by paralleling them along 𝛾, we obtain 𝑛 parallel vector fields {𝐸1 (𝑑), . . . , πΈπ‘›βˆ’1 (𝑑), 𝐸𝑛 (𝑑) = 𝑇}. For any 𝑖 ∈ {1, . . . , 𝑛}, one can get a unique Jacobi vector field along 𝛾 satisfying 𝐽𝑖 (0) = 0, 𝐽𝑖 (π‘Ÿ(π‘ž)) = 𝑒𝑖 . Set π‘Šπ‘– (𝑑) = (𝑑/π‘Ÿ(π‘ž))𝐸𝑖 (𝑑). Then π‘Šπ‘– (0) = 𝐽𝑖 (0) = 0, π‘Šπ‘– (π‘Ÿ(π‘ž)) = 𝐽𝑖 (π‘Ÿ(π‘ž)). Recall that the Hessian of π‘Ÿ is βˆ€π‘‹, π‘Œ ∈ 𝑇π‘₯ 𝑀.

(7)

π‘Ÿ(π‘ž)

0

(9)

1

2

π‘Ÿ (π‘ž)

2

{(

𝑑𝑆 (𝛾 (𝑑) , 𝑇 (𝑑)) + βˆšπ‘ βˆ’ 𝑛) βˆšπ‘ βˆ’ 𝑛

βˆ’ (𝑁 βˆ’ 𝑛)} 𝑑𝑑 ≀ β‹…βˆ«

2

𝑆 (𝛾 (𝑑) , 𝑇 (𝑑)) ) π‘βˆ’π‘›

π‘›βˆ’1 1 βˆ’ π‘Ÿ (π‘ž) π‘Ÿ (π‘ž)2

(6)

pointwise on 𝑀\({𝑝}βˆͺ𝐢𝑒𝑑(𝑝)) and in the sense of distributions on 𝑀\{𝑝}.

π‘›βˆ’1 1 βˆ’ π‘Ÿ (π‘ž) π‘Ÿ (π‘ž)2

{𝑑2 (Ric (𝑇 (𝑑)) + 𝑆 Μ‡ (𝛾 (𝑑) , 𝑇 (𝑑))) + 2𝑑𝑆} 𝑑𝑑

+ 2𝑑𝑆} 𝑑𝑑 =

0

π‘βˆ’1+𝐢 Ξ”π‘Ÿ ≀ π‘Ÿ

} 𝑑𝑑 =

π‘›βˆ’1 1 βˆ’ π‘Ÿ (π‘ž) π‘Ÿ (π‘ž)2

β‹…βˆ«

Ric∞ (𝑉) fl Ric (𝑉) + 𝑆̇ (𝑉) .

2

{(𝑛 βˆ’ 1) βˆ’ Ric (𝑇 (𝑑)) 𝑑2 } 𝑑𝑑 βˆ’ 𝑆 (π‘ž, βˆ‡π‘Ÿ (π‘ž))

𝑑 (𝑑2 𝑆 (𝛾 (𝑑) , 𝑇 (𝑑)))

0

(5)

1 π‘Ÿ (π‘ž)

π‘Ÿ(π‘ž) 1 π‘›βˆ’1 βˆ’ ∫ {Ric (𝑇 (𝑑)) 𝑑2 2 π‘Ÿ (π‘ž) π‘Ÿ (π‘ž) 0

β‹…βˆ«

βˆ€π‘ ∈ (𝑛, ∞) ,

𝐻 (π‘Ÿ) (𝑋, π‘Œ) = π‘‹π‘Œ (π‘Ÿ) βˆ’ π·βˆ‡π‘Ÿ 𝑋 π‘Œ (π‘Ÿ) ,

𝑖=1 𝑛

(4)

Ric𝑁 (𝑉) fl Ric (𝑉) + 𝑆̇ (𝑉) βˆ’

𝑛

𝑖=1

where 𝑉 ∈ 𝑇π‘₯ 𝑀, and 𝛾 : (βˆ’πœ€, πœ€) 󳨀→ 𝑀 is the geodesic with Μ‡ = 𝑉. 𝑆 is called the 𝑆-curvature [2]. Following 𝛾(0) = π‘₯, 𝛾(0) [11], we define 𝑆 Μ‡ (𝑉) fl πΉβˆ’2 (𝑉)

𝑛

󡄨 trπ‘”βˆ‡π‘Ÿ 𝐻 (π‘Ÿ)σ΅„¨σ΅„¨σ΅„¨σ΅„¨π‘ž = βˆ‘π» (π‘Ÿ) (𝑒𝑖 , 𝑒𝑖 ) = βˆ‘πΌπ›Ύ (𝐽𝑖 , 𝐽𝑖 )

(2)

𝜏 is called the distortion of (𝑀, 𝐹, π‘‘πœ‡). To measure the rate of distortion along geodesics, we define 𝑆 (π‘₯, 𝑉) fl

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

1 π‘βˆ’1 βˆ’ 2 π‘Ÿ (π‘ž) π‘Ÿ (π‘ž)

𝑑2 Ric𝑁 (𝑇 (𝑑)) 𝑑𝑑,

where in the fourth expression we use the fact 𝐹(𝑇(𝑑)) = 1. Note that Ric𝑁(π‘₯, βˆ‡π‘Ÿ) β‰₯ βˆ’πΆπ‘Ÿβˆ’2 (π‘₯). This together with (9) yields Ξ”π‘Ÿ|π‘ž ≀

π‘βˆ’1+𝐢 . π‘Ÿ (π‘ž)

(10)

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

Advances in Mathematical Physics

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 π‘βˆ’1+𝐢 (11) , Ξ”π‘Ÿ ≀ π‘Ÿ which yields Ξ”βˆ‡π‘Ÿ π‘Ÿ2 = 2π‘ŸΞ”π‘Ÿ + 2𝐹 (βˆ‡π‘Ÿ)2 ≀ 2 (𝑁 βˆ’ 1 + 𝐢) + 2 = 2 (𝑁 + 𝐢) .

Notice that 𝑑𝐹(𝑝, π‘ž) ≀ πœ‚π‘‘πΉ (π‘ž, 𝑝), βˆ€π‘, π‘ž ∈ 𝑀. From the triangle inequality, one has 𝐡+𝑝 (1) βŠ‚ 𝐡+π‘₯0 (𝑅 + 1) \𝐡+π‘₯0 (𝑅 βˆ’ πœ‚) ,

𝑀

π‘‘πœ‡

2 (𝑁 + 𝐢) vol𝐹 (𝐡+π‘₯0 (𝑅 + 1)) (12)

(13)

𝑀

Let π‘₯0 ∈ πœ•π΅βˆ’π‘ (𝑅) be a given point. Then, 𝑑𝐹(π‘₯0 , 𝑝) = 𝑅. Set 1, 0 ≀ 𝑑 ≀ 𝑅 βˆ’ πœ‚; { { { {𝑅 + 1 βˆ’ 𝑑 πœ“ (𝑑) fl { , 𝑅 βˆ’ πœ‚ ≀ 𝑑 ≀ 𝑅 + 1; { { 1+πœ‚ { 𝑑 β‰₯ 𝑅 + 1, {0,

𝐹 (𝑋)

π‘‹βˆˆπ‘‡π‘€\0 𝐹 (βˆ’π‘‹)

.

β‰₯

2 (𝑅 βˆ’ πœ‚) π‘‘πœ‡ + vol𝐹 (𝐡π‘₯0 (𝑅 + 1) \𝐡+π‘₯0 (𝑅 βˆ’ πœ‚)) 1+πœ‚

β‰₯

2 (𝑅 βˆ’ πœ‚) π‘‘πœ‡ + vol𝐹 𝐡𝑝 (1) . 1+πœ‚

On the other hand, it is not hard to see 𝐡+π‘₯0 (𝑅 + 1) βŠ‚ 𝐡+𝑝 ((πœ‚ + 1)(𝑅 + 1)). Combining this and the formula (20) yields vol𝐹 (𝐡+𝑝 ((πœ‚ + 1) (𝑅 + 1)))

(14)

β‰₯

(15)

βˆ€π‘, π‘ž ∈ 𝑀.

βˆ‡π‘Ÿ

∫ πœ‘Ξ” π‘Ÿ π‘‘πœ‡ = βˆ’ ∫

𝐡+π‘₯0 (𝑅+1)

𝑀

= βˆ’2 ∫

𝐡+π‘₯0 (𝑅+1)

π‘‘πœ‡

vol𝐹 (𝐡+𝑝 (𝑅)) β‰₯

(17)

=

2 ∫ π‘Ÿπ‘‘πœ‡ 1 + πœ‚ 𝐡+π‘₯0 (𝑅+1)\𝐡+π‘₯0 (π‘…βˆ’πœ‚)

β‰₯

2 (𝑅 βˆ’ πœ‚) π‘‘πœ‡ + vol𝐹 (𝐡π‘₯0 (𝑅 + 1) \𝐡+π‘₯0 (𝑅 βˆ’ πœ‚)) , 1+πœ‚

≀ 2 (𝑁 + 𝐢) ∫ πœ‘π‘‘πœ‡ = 2 (𝑁 + 𝐢) ∫

𝐡+π‘₯0 (𝑅+1)

π‘‘πœ‡

= 2 (𝑁 + 𝐢) vol𝐹 (𝐡+π‘₯0 (𝑅 + 1)) .

(22)

(1))) 𝑅.

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 (𝑅 βˆ’ πœ‚) π‘‘πœ‡ + vol𝐹 (𝐡π‘₯0 (𝑅 + 1) \𝐡+π‘₯0 (𝑅 βˆ’ πœ‚)) 1+πœ‚

π‘‘πœ‡

(𝐡+𝑝

Conflicts of Interest

which together with (13) gives

𝐡+π‘₯0 (𝑅+1)

Μƒ (𝑁, 𝐢, πœ‚, volπ‘‘πœ‡ 𝐢 𝐹

Additional Points

πœ“σΈ€  (π‘Ÿ (π‘₯)) π‘ŸπΉ (βˆ‡π‘Ÿ)2 π‘‘πœ‡

≀ 2 (𝑁 + 𝐢) ∫

β‰₯

(𝑅/ (πœ‚ + 1) βˆ’ (1 + πœ‚)) π‘‘πœ‡ + vol𝐹 𝐡𝑝 (1) (𝑁 + 𝐢) (1 + πœ‚)

βˆ‡π‘Ÿ 2

π‘”βˆ‡π‘Ÿ (βˆ‡ πœ‘, βˆ‡ π‘Ÿ ) π‘‘πœ‡

𝑀

(21)

(16)

If πœ‘(π‘₯) = πœ“(π‘Ÿ(π‘₯)), then πœ‘(π‘₯) is a Lipschitz continuous function and supp πœ‘ βŠ‚ 𝐡+π‘₯0 (𝑅 + 1). Since Stokes formula still holds for Lipschitz continuous functions, we have βˆ‡π‘Ÿ 2

(𝑅 βˆ’ πœ‚) π‘‘πœ‡ vol𝐹 𝐡+𝑝 (1) . (𝑁 + 𝐢) (1 + πœ‚)

Replacing (πœ‚ + 1)(𝑅 + 1) by 𝑅, we have

(𝑀, 𝐹) is called reversible if πœ‚ = 1. It is clear that the distance function 𝑑𝐹 of 𝐹 satisfies 𝑑𝐹 (𝑝, π‘ž) ≀ πœ‚π‘‘πΉ (π‘ž, 𝑝) ,

(20)

π‘‘πœ‡

for any 𝑅 > πœ‚, where πœ‚ is the reversibility of 𝐹 defined by (see [12]) πœ‚ = max

(19)

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

Therefore, for any nonnegative function πœ‘ ∈ 𝐢∞ 0 (𝑀), it holds that ∫ πœ‘Ξ”βˆ‡π‘Ÿ π‘Ÿ2 π‘‘πœ‡ ≀ 2 (𝑁 + 𝐢) ∫ πœ‘π‘‘πœ‡.

βˆ€π‘… > πœ‚.

References

πœ‘π‘‘πœ‡ (18)

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4 [3] B. Y. Wu and Y. L. Xin, β€œComparison theorems in Finsler geometry and their applications,” Mathematische Annalen, vol. 337, no. 1, pp. 177–196, 2007. [4] S. Yin, Q. He, and D. X. Zheng, β€œSome comparison theorems and their applications in Finsler geometry,” Journal of Inequalities and Applications, vol. 107, 2014. [5] W. Zhao and Y. Shen, β€œA universal volume comparison theorem for Finsler manifolds and related results,” Canadian Journal of Mathematics, vol. 65, no. 6, pp. 1401–1435, 2013. [6] E. Calabi, β€œOn manifolds with nonnegative Ricci curvature II,” Notices of the American Mathematical Society, vol. 22, 1975. [7] S. T. Yau, β€œSome function-theoretic properties of complete Riemannian manifold and their applications to geometry,” Indiana University Mathematics Journal, vol. 25, no. 7, pp. 659– 670, 1976. [8] J. Cheeger, M. Gromov, and M. Taylor, β€œFinite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds,” Journal of Differential Geometry, vol. 17, no. 1, pp. 15–53, 1982. [9] L. Ma, β€œSome properties of non-compact complete Riemannian manifolds,” Bulletin des Sciences MathΒ΄ematiques, vol. 130, no. 4, pp. 330–336, 2006. [10] B. Y. Wu, β€œVolume form and its applications in Finsler geometry,” Publicationes Mathematicae, vol. 78, no. 3-4, pp. 723–741, 2011. [11] S. Ohta, β€œFinsler interpolation inequalities,” Calculus of Variations and Partial Differential Equations, vol. 36, pp. 211–249, 2009. [12] H.-B. Rademacher, β€œA sphere theorem for non-reversible Finsler metrics,” Mathematische Annalen, vol. 328, no. 3, pp. 373–387, 2004.

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