Hadamard流形中子流形的p-调和函数的刘维尔型定理
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摘要
若Hadamard流形中的完备非紧可定向子流形具有有限全曲率且截面曲率非正,证明了当光滑函数u的Lp范数有限时,任何的p-调和函数一定是一个常数(p≥2).
Let m-dimensional complete non-compact oriented submanifolds in Hadamard manifolds have finite total curvature and non-positive sectional curvature. Further,it is assumed that the first eigenvalue of Laplacian in M is bounded by an appropriate constant. Then,when the norm of Lpof the smooth function u is finite,any p-harmonic function must be a constant( p≥ 2).
Let m-dimensional complete non-compact oriented submanifolds in Hadamard manifolds have finite total curvature and non-positive sectional curvature. Further,it is assumed that the first eigenvalue of Laplacian in M is bounded by an appropriate constant. Then,when the norm of Lpof the smooth function u is finite,any p-harmonic function must be a constant( p≥ 2).
引文
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[10]韩英波,张倩玉,喻丽菊.CC-稳态映射的刘维尔型定理[J].信阳师范学院学报(自然科学版),2017,30(1):22-27.HAN Yingbo,ZHANG Qianyu,YU Liju. Liouville type theorems for CC-stationary maps[J]. Journal of Xinyang NormalUniversity(Natural Science Edition),2017,30(1):22-27.
[11]韩英波,蒋凯歌,张倩玉.具有势函数的拟-F-调和映射的若干结果[J].信阳师范学院学报(自然科学版),2018,31(1):5-10.HAN Yingbo,JIANG Kaige,ZHANG Qianyu. Some results for quasi-F-harmonic maps with potential[J]. Journal ofXinyang Normal University(Natural Science Edition),2018,31(1):5-10.
[12] NAKAUCHI N. A Liouville type theorem for p-harmonic maps[J]. Osaka Journal of Mathematics,1998,35(2):303-312.
[13] HOFFMAN D,SPRUCK J. Sobolev and isoperimetric inequalities for Riemannian submanifolds[J]. Communications onPure and Applied Mathematics,1974,27(6):715-727.
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[15] HAN Y B,FENG S X. A Liouville type theorem for p-haromonic functions on minimal submanifolds in Rn+m[J]. MatematiqkiVesnik,2013,65(4):494-498.
[2] SCHOEN R,YAU S T. Harmonic maps and the topology of the stable hypersurfaces and manifolds with non-negatie Ricci cur-vature[J]. Commentarii Mathematici Helvetici,1976,51(1):33-341.
[3] NI L. Gap theorems for minimal submanifolds in Rn+1[J]. Communications in Analysis and Geometry,2001,9(3):641-656.
[4] SEO Keomkyo. Minimal submanifolds with small total scalar curvature in Euclidean space[J]. Kodai Mathematical Journal,2008,31(1):113-119.
[5] LI P,TAM L F. Harmonic functions and the structure of complete manifolds[J]. Journal of Differential Geometry,1992,35(2):359-383.
[6] FU H P,XU H W. Total curvauture and L2harmonic-forms on complete submanifolds in space forms[J]. Geometriae Dedi-cate,2010,144(1):129-140.
[7] CAVALCANTE M P,MIRANDOLA H,VITRIO F. L2harmonic 1-forms on submanifolds with finite total curvature[J].Journal Geometric Analysis,2014,24(1):205-222.
[8] BAIRD P,GUDMUNDSSON S. p-harmonic maps and submanifolds[J]. Mathematische Annalen,1992,294(1):611-624.
[9]韩英波,方联银,李静.带位势的弱F-调和映照的单调公式[J].信阳师范学院学报(自然科学版),2016,29(2):165-170.HAN Yingbo,FANG Lianyin,LI Jing. Monotonicity formulas of weakly F-harmonic map with potential[J]. Journal ofXinyang Normal University(Natural Science Edition),2016,29(2):165-170.
[10]韩英波,张倩玉,喻丽菊.CC-稳态映射的刘维尔型定理[J].信阳师范学院学报(自然科学版),2017,30(1):22-27.HAN Yingbo,ZHANG Qianyu,YU Liju. Liouville type theorems for CC-stationary maps[J]. Journal of Xinyang NormalUniversity(Natural Science Edition),2017,30(1):22-27.
[11]韩英波,蒋凯歌,张倩玉.具有势函数的拟-F-调和映射的若干结果[J].信阳师范学院学报(自然科学版),2018,31(1):5-10.HAN Yingbo,JIANG Kaige,ZHANG Qianyu. Some results for quasi-F-harmonic maps with potential[J]. Journal ofXinyang Normal University(Natural Science Edition),2018,31(1):5-10.
[12] NAKAUCHI N. A Liouville type theorem for p-harmonic maps[J]. Osaka Journal of Mathematics,1998,35(2):303-312.
[13] HOFFMAN D,SPRUCK J. Sobolev and isoperimetric inequalities for Riemannian submanifolds[J]. Communications onPure and Applied Mathematics,1974,27(6):715-727.
[14] SHIOHAMA K,XU H. The topological sphere for complete submanifolds[J]. Compostio Mathematica,1997,107(2):221-232.
[15] HAN Y B,FENG S X. A Liouville type theorem for p-haromonic functions on minimal submanifolds in Rn+m[J]. MatematiqkiVesnik,2013,65(4):494-498.