AN ESTIMATE FOR THE MEAN CURVATURE OF SUBMANIFOLDS CONTAINED IN A HOROBALL

doi:10.1016/S0252-9602(13)60104-6

数学物理学报(英文版) ›› 2013, Vol. 33 ›› Issue (6): 1561-1570.doi: 10.1016/S0252-9602(13)60104-6

AN ESTIMATE FOR THE MEAN CURVATURE OF SUBMANIFOLDS CONTAINED IN A HOROBALL

邱红兵

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

收稿日期:2012-07-31 出版日期:2013-11-20 发布日期:2013-11-20
基金资助:
The research is partially supported by the National Natural Science Foundation of China (11126189, 11171259), Specialized Research Fund for the Doctoral Program of Higher Education (20120141120058), China Postdoctoral Science Foundation Funded Project (20110491212) and the Fundamental Research Funds for the Central Universities (2042011111054).

AN ESTIMATE FOR THE MEAN CURVATURE OF SUBMANIFOLDS CONTAINED IN A HOROBALL

QIU Hong-Bing

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received:2012-07-31 Online:2013-11-20 Published:2013-11-20
Supported by:
The research is partially supported by the National Natural Science Foundation of China (11126189, 11171259), Specialized Research Fund for the Doctoral Program of Higher Education (20120141120058), China Postdoctoral Science Foundation Funded Project (20110491212) and the Fundamental Research Funds for the Central Universities (2042011111054).

摘要/Abstract

摘要：

We obtain the Omori-Yau maximum principle on complete properly immersed submanifolds with the mean curvature satisfying certain condition in complete Riemannian manifolds whose radial sectional curvature satisfies some decay condition, which generalizes our previous results in [17]. Using this generalized maximum principle, we give an estimate
on the mean curvature of properly immersed submanifolds in Hⁿ ×R^? with the image under the projection on Hn contained in a horoball and the corresponding situation in hyperbolic space. We also give other applications of the generalized maximum principle.

关键词: Omori-Yau maximum principle, proper, mean curvature, horoball, hyper-bolic space

Abstract:

Key words: Omori-Yau maximum principle, proper, mean curvature, horoball, hyper-bolic space

中图分类号:

53C40

邱红兵. AN ESTIMATE FOR THE MEAN CURVATURE OF SUBMANIFOLDS CONTAINED IN A HOROBALL[J]. 数学物理学报(英文版), 2013, 33(6): 1561-1570.

QIU Hong-Bing. AN ESTIMATE FOR THE MEAN CURVATURE OF SUBMANIFOLDS CONTAINED IN A HOROBALL[J]. Acta mathematica scientia,Series B, 2013, 33(6): 1561-1570.

参考文献

[1] Alias L J, Bessa G P, Dajczer M. The mean curvature of cylindrically bounded submanifolds. Math Ann, 2009, 345: 367–376

[2] Calabi E. Problems in differential geometry//Kobayashi S, Eells J Jr, eds. Proceedings of the United States-Japan Seminar in Differential Geometry, Kyoto, Japan, 1965. Tokyo: Nippon Hyoronsha, 1966

[3] Chen Q, Xin Y L. A generalized maximum principle and its applications in geometry. Amer J Math, 1992, 114: 355–366

[4] Cheng S Y, Yau S T. Differential equations on Riemannian manifolds and their geometric applications. Comm Pure Appl Math, 1975, 28: 333–354

[5] Chern S S. The geometry of G-structures. Bull Amer Math Soc, 1966, 72: 167–219

[6] Colding T H, Minicozzi II W P. The Calabi-Yau conjectures for embedded surfaces. Ann Math, 2008, 167: 211–243

[7] Huang X G. The maximum principle on complete Riemannian manifolds (in Chinese). Chin Ann Math, 1993, 14A(2): 175–185

[8] Jorge L, Koutroufiotis D. An estimate for the curvature of bounded submanifolds. Amer J Math, 1980, 103: 711–725

[9] Jorge L, Xavier F. A complete minimal surface in R3 between two parallel planes. Ann Math, 1980, 112: 203–206

[10] Jorge L, Xavier F. An inequality between the exterior diameter and mean curvature of bounded immersions. Math Z, 1981, 178: 72–82

[11] Lluch A. Isometric immersions in the hyperbolic space with their image contained in a horoball. Glasgow Math J, 2001, 43: 1–8

[12] Lima B P, Pessoa L F. On the Omori-Yau maximum principle and geometric applications. 2012, arXiv: math.DG/1201.1675v1

[13] Nadirashvili N. Hadamard’s and Calabi-Yau’s conjectures on negatively curved and minimal surfaces. Invent Math, 1996, 126: 457–465

[14] O’Neill B. Semi-Riemannian Geometry. Academic Press, 1983

[15] Omori H. Isometric immersion of Riemannian manifolds. J Math Soc Japan, 1967, 19: 205–214

[16] Pigola S, Rigoli M, Setti A. Maximum principle on Riemannian manifolds and applications. Mem Amer Math Soc, 2005, 174(822)

[17] Qiu H, Xin Y L. Proper manifolds in product manifolds. Chin Ann Math, 2012, 33B(1): 1–16

[18] Xin Y L. Geometry of Harmonic Maps. Birkh¨auser, 1994

[19] Xu S, Ni Y. Submanifolds of product Riemannian manifold. Acta Math Sci, 2000, 20B(2): 213–218

[20] Yau S T. Harmonic function on complete Riemannian manifolds. Comm Pure Appl Math, 1975, 28: 201–228

AN ESTIMATE FOR THE MEAN CURVATURE OF SUBMANIFOLDS CONTAINED IN A HOROBALL

AN ESTIMATE FOR THE MEAN CURVATURE OF SUBMANIFOLDS CONTAINED IN A HOROBALL

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 10

[1]	范庆斋, 方小春. CROSSED PRODUCTS BY FINITE GROUP ACTIONS WITH CERTAIN TRACIAL ROKHLIN PROPERTY[J]. 数学物理学报(英文版), 2018, 38(3): 829-842.
[2]	王志明. EULER SCHEME AND MEASURABLE FLOWS FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH NON-LIPSCHITZ COEFFICIENTS[J]. 数学物理学报(英文版), 2018, 38(1): 157-168.
[3]	马如云, 高红亮. MULTIPLE POSITIVE SOLUTIONS FOR A CLASS OF SEMIPOSITONE NEUMANN PROBLEMS WITH SINGULAR φ-LAPLACIAN[J]. 数学物理学报(英文版), 2017, 37(5): 1472-1482.
[4]	何春蕾, 黄守军, 邢晓敏. SELF-SIMILAR SOLUTIONS TO THE HYPERBOLIC MEAN CURVATURE FLOW[J]. 数学物理学报(英文版), 2017, 37(3): 657-667.
[5]	Mohamed AMOUCH, Youness FAOUZI. HYPERCYCLICITY, SUPERCYCLICITY AND GENERALIZED RAKO?EVI?'S PROPERTY[J]. 数学物理学报(英文版), 2017, 37(2): 503-510.
[6]	王静, 张银山. RIGIDITY OF COMPACT SURFACES IN HOMOGENEOUS 3-MANIFOLDS WITH CONSTANT MEAN CURVATURE[J]. 数学物理学报(英文版), 2016, 36(6): 1609-1618.
[7]	谌稳固, 李亚玲. STABLE RECOVERY OF SIGNALS WITH THE HIGH ORDER D-RIP CONDITION[J]. 数学物理学报(英文版), 2016, 36(6): 1721-1730.
[8]	Kamal OULD BOUH. EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR A HARMONIC EQUATION WITH CRITICAL NONLINEARITY[J]. 数学物理学报(英文版), 2016, 36(5): 1305-1316.
[9]	黄煜可, 文志英. KERNEL WORDS AND GAP SEQUENCE OF THE TRIBONACCI SEQUENCE[J]. 数学物理学报(英文版), 2016, 36(1): 173-194.
[10]	鲍玲鑫. ON WEAK FILTER CONVERGENCE AND THE RADON-RIESZ TYPE THEOREM[J]. 数学物理学报(英文版), 2016, 36(1): 215-219.
[11]	刘芳. SOLUTIONS TO A CLASS OF PARABOLIC INHOMOGENEOUS NORMALIZED p-LAPLACE EQUATIONS[J]. 数学物理学报(英文版), 2015, 35(2): 477-494.
[12]	Moosa GABELEH. BEST PROXIMITY POINT THEOREMS FOR SINGLE- AND SET-VALUED NON-SELF MAPPINGS[J]. 数学物理学报(英文版), 2014, 34(5): 1661-1669.
[13]	Dae-Woong LEE. ON INVERSES AND ALGEBRAIC LOOPS OF CO-H-SPACES[J]. 数学物理学报(英文版), 2014, 34(4): 1193-1211.
[14]	罗振东, 李宏. A POD REDUCED-ORDER SPDMFE EXTRAPOLATING ALGORITHM FOR HYPERBOLIC EQUATIONS[J]. 数学物理学报(英文版), 2014, 34(3): 872-890.
[15]	赵月旭. ASYMPTOTIC PROPERTIES OF THE MOMENT CONVERGENCE FOR NA SEQUENCES[J]. 数学物理学报(英文版), 2014, 34(2): 301-312.