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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 張樹城(Shu-Cheng Chang) | |
| dc.contributor.author | Li-Chung Yu | en |
| dc.contributor.author | 游禮中 | zh_TW |
| dc.date.accessioned | 2021-06-15T16:10:20Z | - |
| dc.date.available | 2015-08-25 | |
| dc.date.copyright | 2015-08-25 | |
| dc.date.issued | 2015 | |
| dc.date.submitted | 2015-08-18 | |
| dc.identifier.citation | [1] A. Agrachev and W.-Y. Lee, Bishop and Laplacian Comparison Theorems on Three Dimensional Contact Sub-Riemannian Manifolds with Symmetry, to appear in JGEA.
[2] D. Bakry and M. Emery, Diffusion hypercontractives, Sèm. Prob. XIX, Lect. Notes in Maths. 1123 (1985) 177-206. [3] S.-C. Chang and H.-L. Chiu, Nonnegativity of CR Paneitz operator and its Application to the CR Obata's Theorem in a Pseudohermitian (2n+1)-Manifold, JGA, vol 19 (2009), 261-287. [4] S.-C. Chang, T.-J. Kuo, and S.-H. Lai, Li-Yau Gradient Estimate and Entropy Formulae for the heat equation in a Closed Pseudohermitian 3-manifold, J. Differential Geom. 89 (2011), 185-216. [5] S.-C. Chang, Jingzhu Tie and C.-T. Wu, Subgradient Estimate and Liouville-type Theorems for the CR Heat Equation on Heisenberg groups Hn, Asian J. Math., Vol. 14, No. 1 (2010), 041-072 [6] H.-D. Cao and S.-T. Yau, Gradient Estimates, Harnack Inequalities and Estimates for Heat Kernels of the Sum of Squares of Vector Fields, Math. Z. 211 (1992), 485-504. [7] C Fefferman and K. Hirachi, Ambient Metric Construction of Q-Curvature in Conformal and CR Geometries, Math. Res. Lett., 10, No. 5-6 (2003), 819-831. [8] C. R. Graham and J. M. Lee, Smooth Solutions of Degenerate Laplacians on Strictly Pseudoconvex Domains, Duke Math. J., 57 (1988), 697-720. [9] R.-S. Hamilton, Three-Manifolds with Positive Ricci Curvature, J. Diff. Geom. 17 (1982), 255-306. [10] K. Hirachi, Scalar Pseudo-hermitian Invariants and the SzegöKernel on 3-dimensional CR Manifold, Lecture Notes in Pure and Appl. Math. 143, pp. 67-76 Dekker, 1992. [11] J. M. Lee, Pseudo-Einstein Structure on CR Manifolds, Amer. J. Math. 110 (1988), 157-178. [12] J. M. Lee, The Fefferman Metric and Pseudohermitian Invariants, Trans. A.M.S. 296 (1986), 411-429. [13] P. Li, Lecture on Harmonic Functions, UCI, 2014. [14] X.-D. Li, Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds, J. Math. Pures Appl. 84 (2005), 1295-1361. [15] X.-D. Li, Perelman's Entropy Formula for the Witten Laplacian on Riemannian Manifolds via Bakry-Emery Ricci Curvature, preprint. [16] P. Li and S.-T. Yau, On the Parabolic Kernel of the Schrödinger Operator, Acta Math.156 (1985), 153-201. [17] Q. H. Ruan, Bakry-Emery Curvature Operator and Ricci Flow, Potential Analysis, 25 (2006), No. 4, 399-406. [18] R. Strichartz, Sub-Riemannian geometry, J. Differential Geom. 24 (1986) 221-263. [19] R. Schoen and S.-T. Yau, Lectures on Differentail Geometry, International Press, 1994. [20] N. Tanaka, A Differential Geometric Study on Strongly Pseudoconvex Manifolds, Lectures in Mathematics, Kyoto University, Kinokuniya Book Store 9. [21] S. M. Webster, Pseudo-Hermitian Structures on a Real Hypersurface, J. Differential Geom. 13 No.1 (1978) 25-41. [22] L.-M. Wu, Uniqueness of Nelson's Diffusions, Probab. Theory and Related Fields, 114 (1999), 549-585. [23] S.-T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201-228. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/52252 | - |
| dc.description.abstract | 這篇論文主要是模仿丘成桐教授在1975年對黎曼流型上的L同調函數作梯度估計的方法，進一步的引用在加權柯西黎曼流型上面。由於考慮的為加權流型，因此考慮的曲率將從柯西黎曼里奇曲率，調整成考慮Bakry-Emery 里奇曲率。經由論文內的計算我們可得知，當這個曲率具有下界時，L同調正函數的梯度將會有上界。更進一步來說，若此曲率的下界為零時，則此函數將會是常數函數。換句話說，當此曲率下界為零，我們將會得到加權柯西黎曼流型上的劉維爾定理。 | zh_TW |
| dc.description.abstract | In this paper, we modify Yau's method to discuss a gradient estimate of a nonnegative L-pseudoharmonic function on a oriented, complete, pseudohermitian manifold which satisfies Witten-sub-Laplacian comparison property. Since the manifold we considered in this paper is weighted manifold, the curvature we consider is not only Ricci curvature but Bakry-Emery Ricci curvature Ric_m,n (L). At the end of this paper, we can get that when the form 2Ric_m,n (L) - Tor(L) is bounded below, any gradient estimate of a nonnegative L-pseudoharmonic function is bounded. Moreover, we can then deduce Liouville property on such manifold with curvature satisfies 2Ric_m,n (L) > Tor(L). | en |
| dc.description.provenance | Made available in DSpace on 2021-06-15T16:10:20Z (GMT). No. of bitstreams: 1 ntu-104-R02221025-1.pdf: 1436239 bytes, checksum: 934b63c1a901a78d7e9d36432535f758 (MD5) Previous issue date: 2015 | en |
| dc.description.tableofcontents | 誌謝 i
中文摘要 ii ABSTRACT iii CONTENTS iv LIST OF FIGURES vi LIST OF TABLES vii Chapter 1 Introduction 1 Chapter 2 Preliminary 3 Chapter 3 CR Analogue of Yau's Gradient Estimate 7 REFERENCE 22 | |
| dc.language.iso | en | |
| dc.subject | 梯度估計 | zh_TW |
| dc.subject | 加權流型、柯西黎曼流型 | zh_TW |
| dc.subject | Bakry-Emery 里奇曲率 | zh_TW |
| dc.subject | L同調函數 | zh_TW |
| dc.subject | gradient estimate | en |
| dc.subject | weighted manifold | en |
| dc.subject | CR manifold | en |
| dc.subject | Bakry-Emery Ricci curvature | en |
| dc.subject | L-harmonicfunction | en |
| dc.title | 班克明-愛茉莉擬赫米遜里奇曲率下完備擬赫米遜流型上L-擬調和函數的梯度估計及劉維爾性質 | zh_TW |
| dc.title | Gradient Estimate and Liouville Property of L-pseudoharmonic Functions on a Complete Pseudohermitian Manifold with Bakry-Emery Pseudohermitian Ricci Curvature | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 103-2 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 王藹農(Ai-Nung Wang),陳瑞堂(Jui-Tang Chen) | |
| dc.subject.keyword | 加權流型、柯西黎曼流型,Bakry-Emery 里奇曲率,L同調函數,梯度估計, | zh_TW |
| dc.subject.keyword | weighted manifold,CR manifold,Bakry-Emery Ricci curvature,L-harmonicfunction,gradient estimate, | en |
| dc.relation.page | 23 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2015-08-19 | |
| dc.contributor.author-college | 理學院 | zh_TW |
| dc.contributor.author-dept | 數學研究所 | zh_TW |
| 顯示於系所單位： | 數學系 | |
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