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研究生:王一珊
研究生(外文):Wang, Yi-Shan
論文名稱:完備流形上的調和函數空間
論文名稱(外文):A note on the space of harmonic functions
指導教授:宋瓊珠
指導教授(外文):Sung, Chiung-Jue
學位類別:碩士
校院名稱:國立清華大學
系所名稱:數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2012
畢業學年度:100
論文頁數:44
中文關鍵詞:調和函數
外文關鍵詞:Harmonic functionGradient estimateEnd
相關次數:
  • 被引用被引用:0
  • 點閱點閱:148
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  • 下載下載:10
  • 收藏至我的研究室書目清單書目收藏:0
In this note, we introduce various spaces of harmonic functions on a complete Riemannian manifold, and present the study of how the space of polynomial growth harmonic functions on a complete manifold can be reduced to each end of the manifold. Finally, the finite dimensionality for such spaces on a manifold is established.
1. Introduction
2. Prelimilaries
3. Gradient estimate
4. Dimension identities
5. Finite dimensionality
Reference
References

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

[3] S. Y. Cheng, Finite dimensionality of the space of bounded and harmonic functions, preprint.
[4] S. Y. Cheng, Liouville theorem for harmonic maps, Proc. Symp. Pure. Math. 36 (1980), 147–151 [5] T. Cloding and W. Minicozzi, Harmonic Functions on Manifolds, Ann. Math. 146 (1997), 725–747 [6] T. Cloding and W. Minicozzi, On Function Theory on Spaces with a Lower Ricci Curvature Bound,
Math. Res. Lett. 3 (1996), 241–246

[7] T. Cloding and W. Minicozzi, Generalized Liouville properties of manifolds, Math. Res. Lett. 3
(1996), 723–729

[8] T. Cloding and W. Minicozzi, Harmonic functions with polynomial growth, J. Differential Geom. 46
(1997), 1–77

[9] T. Cloding and W. Minicozzi, Large scale behavior of kernels of Schrdinger operators, Amer. J.
Math. 119 (1997), 1355–1398

[10] T. Cloding and W. Minicozzi, Liouville theorems for harmonic sections and applications, Comm.
Pure Appl. Math. 51 (1998), 113–138

[11] T. Cloding and W. Minicozzi, Weyl type bounds for harmonic functions, Invent. Math. 131 (1998),
257–298

[12] E. Constantin and N. H. Pavel, Green function of the Laplacian for the Neumann Problem in Rn ,
Libertas Mathematica XXX (2010), 57–69

[13] H. Donnelly, Bounded harmonic functions and positive Ricci curvature, Math. Z. 191 (1986), 559–
565

[14] P. Li and J. Wang, Complete manifolds with positive spectrum, II. J. Differential Geom. 62 (2002),
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[15] P. Li and J. Wang, Mean value inequality, Indiana Univ. Math. J. 48 (1999), no 4, 1257–1283

[16] P. Li and J. Wang, Polynomial growth solutions of uniformly elliptic operators of nondivergence form, Proc. AMS 129 (2001), 3691–3699

[17] P. Li and L. F. Tam, Symmetric Green’s Function on Complete Manifold, Amer. J. Math. 109
(1987), 1129–1154

[18] P. Li and L. F. Tam, Linear growth harmonic functions on a complete manifold, J. Differential
Geom. 29 (1989), 421–425

[19] P. Li and L. F. Tam, Harmonic functions and the structure of complete manifolds, J. Differential
Geom. 35 (1992), 359–383

[20] P. Li and L. F. Tam, Positive harmonic functions on complete manifolds with non-negative curvature outside a compact set, Ann. Math. 125 (1987), 171–107

[21] P. Li, Harmonic sections of polynomial growth, Math. Res. Lett. 4 (1997), 35–44 [22] P. Li, Harmonic functions and applications to complete manifolds, Lecture notes
[23] Ovidiu Munteanu, On the gradient estimate of Cheng and Yau, Proc. Amer. Math. Soc. 140 (2012)
no 4, 1437–1443

[24] Ovidiu Munteanu and Jiaping Wang, Smooth metric measure spaces with nonnegative curvature,
Comm. Anal. Geom. 19 (2011) no 3, 451–486

[25] M. Nakai, On evans potential, Proc. Japan. Acad. 38 (1962), 624–629

[26] C. J. Sung, L. F Tam ,and J. Wang, Space of harmonic functions, J. London Math Soc. (2) 61
(2000), 789–806

[27] C. J. Sung Harmonic functions under quasi-isometry, J. Geom. Analysis (8) 1 (1998), 143–161

[28] R. Schoen and S. T. Yau, Lectures on differential geometry, (Cambridge, MA : International Press,
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[30] S. T. Yau, Harmonic function on complete manifold, Comm. Pure Appl. Math. 28 (1975), 201–228

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