THE RELATIVE VOLUME FUNCTION AND THE CAPACITY OF SPHERE ON ASYMPTOTICALLY HYPERBOLIC MANIFOLDS

doi:10.1007/s10473-025-0301-9

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (3): 755-770.doi: 10.1007/s10473-025-0301-9

• • 下一篇

THE RELATIVE VOLUME FUNCTION AND THE CAPACITY OF SPHERE ON ASYMPTOTICALLY HYPERBOLIC MANIFOLDS

Xiaoshang JIN

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

收稿日期:2023-12-27 修回日期:2024-05-05 出版日期:2025-05-25 发布日期:2025-09-30

THE RELATIVE VOLUME FUNCTION AND THE CAPACITY OF SPHERE ON ASYMPTOTICALLY HYPERBOLIC MANIFOLDS

Xiaoshang JIN

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Received:2023-12-27 Revised:2024-05-05 Online:2025-05-25 Published:2025-09-30
About author:Xiaoshang JIN, E-mail: jinxs@hust.edu.cn
Supported by:
NSFC (Grant No. 12201225).

摘要/Abstract

摘要： Following the work of Li-Shi-Qing, we propose the definition of the relative volume function for an AH manifold. It is not a constant function in general and we study the regularity of this function. We use this function to provide an accurate characterization of the height of the geodesic defining function for the AH manifold with a given boundary metric. Furthermore, it is shown that such functions are uniformly bounded from below at infinity and the bound only depends on the dimension. In the end, we apply this function to study the capacity of balls in AH manifolds and demonstrate that the "relative $p$-capacity function" coincides with the relative volume function under appropriate curvature conditions.

关键词: conformally compact, relative volume, geodesic defining function, capacity

Abstract: Following the work of Li-Shi-Qing, we propose the definition of the relative volume function for an AH manifold. It is not a constant function in general and we study the regularity of this function. We use this function to provide an accurate characterization of the height of the geodesic defining function for the AH manifold with a given boundary metric. Furthermore, it is shown that such functions are uniformly bounded from below at infinity and the bound only depends on the dimension. In the end, we apply this function to study the capacity of balls in AH manifolds and demonstrate that the "relative $p$-capacity function" coincides with the relative volume function under appropriate curvature conditions.

Key words: conformally compact, relative volume, geodesic defining function, capacity

中图分类号:

53C18

Xiaoshang JIN. THE RELATIVE VOLUME FUNCTION AND THE CAPACITY OF SPHERE ON ASYMPTOTICALLY HYPERBOLIC MANIFOLDS[J]. 数学物理学报(英文版), 2025, 45(3): 755-770.

Xiaoshang JIN. THE RELATIVE VOLUME FUNCTION AND THE CAPACITY OF SPHERE ON ASYMPTOTICALLY HYPERBOLIC MANIFOLDS[J]. Acta mathematica scientia,Series B, 2025, 45(3): 755-770.

参考文献

[1] Bray H, Miao P. On the capacity of surfaces in manifolds with nonnegative scalar curvature. Inventiones Mathematicae, 2008, 172(3): 459-475
[2] Chruściel P T, Delay E, Lee J M, Skinner D N. Boundary regularity of conformally compact Einstein metrics. Journal of Differential Geometry, 2005, 69(1): 111-136
[3] Dutta S, Javaheri M. Rigidity of conformally compact manifolds with the round sphere as the conformal infinity. Advances in Mathematics, 2020, 224(2): 525-538
[4] Graham C R.Volume and area renormalizations for conformally compact Einstein metrics//Slovák J, $\check{\rm c}$adek M. 19th Winter School "Geometry and Physics". Palermo: Circolo Matematico di Palermo, 2000: 31-42
[5] Graham C R, Lee J M. Einstein metrics with prescribed conformal infinity on the ball. Advances in Mathematics, 1991, 87(2): 186-225
[6] Grigor'yan A. Isoperimetric inequalities and capacities on Riemannian manifolds//Rossmann J, Taká$\check{\rm c}$ P, Wildenhain G. The Maz'ya Anniversary Collection. Basel: Birkhäuser, 1999: 139-153
[7] Hijazi O, Montiel S, Raulot S. The Cheeger constant of an asymptotically locally hyperbolic manifold and the yamabe type of its conformal infinity. Communications in Mathematical Physics, 2020, 374(2): 873-890
[8] Jin X. Finite boundary regularity for conformally compact Einstein manifolds of dimension 4. The Journal of Geometric Analysis, 2021, 31(4): 4004-4023
[9] Lee J M. The spectrum of an asymptotically hyperbolic Einstein manifold.Communications in Analysis and Geometry, 1995, 3(2): 253-271
[10] Li G, Qing J, Shi Y. Gap phenomena and curvature estimates for conformally compact Einstein manifolds. Transactions of the American Mathematical Society, 2017, 369(6): 4385-4413
[11] Maldacena J. The large-n limit of superconformal field theories and supergravity. International Journal of Theoretical Physics, 1999, 38(4): 1113-1133
[12] Maz'ya V. Sobolev Spaces. Berlin: Springer, 2013
[13] Mazzeo R. The hodge cohomology of a conformally compact metric. Journal of Differential Geometry, 1988, 28(2): 309-339
[14] Petersen P. Riemannian Geometry. Cham: Springer, 2006
[15] Pólya G.Isoperimetric Inequalities in Mathematical Physics. Princeton: Princeton University Press, 1951
[16] Shi Y, Tian G. Rigidity of asymptotically hyperbolic manifolds. Communications in Mathematical Physics, 2005, 259(3): 545-559

THE RELATIVE VOLUME FUNCTION AND THE CAPACITY OF SPHERE ON ASYMPTOTICALLY HYPERBOLIC MANIFOLDS

THE RELATIVE VOLUME FUNCTION AND THE CAPACITY OF SPHERE ON ASYMPTOTICALLY HYPERBOLIC MANIFOLDS

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 14

Metrics

本文评价

推荐阅读 0

[1]	Zhenlong Chen, Weijie Yuan. MULTIPLE INTERSECTIONS OF SPACE-TIME ANISOTROPIC GAUSSIAN FIELDS^*[J]. 数学物理学报(英文版), 2024, 44(1): 275-294.
[2]	季乐文, 杨志辉. THE DISCRETE ORLICZ-MINKOWSKI PROBLEM FOR $p$-CAPACITY[J]. 数学物理学报(英文版), 2022, 42(4): 1403-1413.
[3]	王军, 陈振龙. HITTING PROBABILITIES AND INTERSECTIONS OF TIME-SPACE ANISOTROPIC RANDOM FIELDS[J]. 数学物理学报(英文版), 2022, 42(2): 653-670.
[4]	张立新. STRONG LIMIT THEOREMS FOR EXTENDED INDEPENDENT RANDOM VARIABLES AND EXTENDED NEGATIVELY DEPENDENT RANDOM VARIABLES UNDER SUB-LINEAR EXPECTATIONS[J]. 数学物理学报(英文版), 2022, 42(2): 467-490.
[5]	Fredj ELKHADHRA. LELONG-DEMAILLY NUMBERS IN TERMS OF CAPACITY AND WEAK CONVERGENCE FOR CLOSED POSITIVE CURRENTS[J]. 数学物理学报(英文版), 2013, 33(6): 1652-1666.
[6]	Veli Shakhmurov. ABSTRACT CAPACITY OF REGIONS AND COMPACT EMBEDDING WITH APPLICATIONS[J]. 数学物理学报(英文版), 2011, 31(1): 49-67.
[7]	陈振龙, 李慧琼. POLAR SETS OF MULTIPARAMETER BIFRACTIONAL BROWNIAN SHEETS[J]. 数学物理学报(英文版), 2010, 30(3): 857-872.
[8]	陈振龙. GENERALIZED BROWNIAN SHEET IMAGES AND BESSEL-RIESZ CAPACITY[J]. 数学物理学报(英文版), 2007, 27(1): 77-91.
[9]	杨超; 张建中. ON THE BOTTLENECK CAPACITY EXPANSION PROBLEMS ON NETWORKS[J]. 数学物理学报(英文版), 2006, 26(2): 202-208.
[10]	卢广存. THE WEINSTEIN CONJECTURE ON C1-SMOOTH HYPERSURFACE OF CONTACT TYPE[J]. 数学物理学报(英文版), 2002, 22(4): 557-563.
[11]	杨超, 陈学旗. AN INVERSE MAXIMUM CAPACITY PATH PROBLEM WITH LOWER BOUND CONSTRAINTS[J]. 数学物理学报(英文版), 2002, 22(2): 207-212.
[12]	杨超, 刘静林. A CAPACITY EXPANSION PROBLEM WITH BUDGET CONSTRAINT AND BOTTLENECK LIMITATION[J]. 数学物理学报(英文版), 2001, 21(3): 428-432.
[13]	骆顺龙. COLLISION AND MEETING OF STABLE PROCESSES[J]. 数学物理学报(英文版), 1996, 16(3): 271-278.
[14]	刘禄勤. THE EQUILIBRIUM PROBLEM AND CAPACITY FOR JUMP MARKOV PROCESSES[J]. 数学物理学报(英文版), 1995, 15(1): 15-30.