博碩士論文 972401001 詳細資訊




以作者查詢圖書館館藏 以作者查詢臺灣博碩士 以作者查詢全國書目 勘誤回報 、線上人數:2 、訪客IP:3.233.217.242
姓名 賴馨華(Sin-Hua Lai)  查詢紙本館藏   畢業系所 數學系
論文名稱 擬埃爾米特流形上Perelman 的熵公式和海森堡群上的基本定理
(Perelman’s Entropy Formula on Pseudohermitian Manifolds and Fundamental Theorem on Heisenberg Groups)
相關論文
★ 一些橢圓算子的研究★ 從高斯-波涅與黎曼-羅赫定理看指標定理
★ Perturbation Theory for Linear Operators★ 超曲面的基本定理
★ Partially Volume-Preserving Normal Form for Strongly Pseudoconvex Real Hypersurfaces in 2-dimensional Complex Space★ Subdivergence theorem on Heisenberg group
檔案 [Endnote RIS 格式]    [Bibtex 格式]    [相關文章]   [文章引用]   [完整記錄]   [館藏目錄]   [檢視]  [下載]
  1. 本電子論文使用權限為同意立即開放。
  2. 已達開放權限電子全文僅授權使用者為學術研究之目的,進行個人非營利性質之檢索、閱讀、列印。
  3. 請遵守中華民國著作權法之相關規定,切勿任意重製、散佈、改作、轉貼、播送,以免觸法。

摘要(中) 這篇文章分成二個部分:第一個部分我們藉由擬埃爾米特流形的Bakry-Emery瑞奇曲率來研究擬埃爾米特流形上與Witten拉普拉斯聯繫在一起的柯西-黎曼熱方程的Perelman’s W-熵公式。第二部分我們建立在海森堡群中Legendrian子流形的基本定理。
第二章,我們導出在(2n+1)維度閉的擬埃爾米特流形與Witten拉普拉斯聯繫在一起的柯西-黎曼熱方程的次梯度估計。對於此次梯度估計的應用,我們得到柯西-黎曼熱方程的Perelman型式的熵公式和與Witten拉普拉斯聯繫在一起的柯西-黎曼熱方程的Perelman型式的熵公式。
第三章,PSH(n)是由n維海森堡群上所有擬埃爾米特變換所形成的李群。我們得到PSH(n)的群表示。除此之外,我們討論這個矩陣李群PSH(n)中的任一元素如何做為在齊次空間PSH(n)/U(n)上的一組標架。因此我們從Maurer-Cantan form可以立即得到活動標架公式。
第四章,我們利用Élie Cartan的活動標架法、李群理論得到海森堡群中的Legendrian子流形的基本定理。我們令Σ是一個n維可定向的曲面,並且做為在海森堡群中的Legendrian子流形。對於任意在Σ裡的totally real point,我們計算Darboux 導數而得到integrability條件。於是我們可以証明對任意的n維黎曼流形如果滿足integrability條件,那麼此黎曼流形就可以局部嵌入到n維的海森堡群裡,做為海森堡群中的Legendrian子流形。
摘要(英) In this thesis, we study Perelman’s W-Entropy formula for the CR heat equation associated with the Witten Laplacian on pseudohermitian manifolds via the Bakry-Emery Ricci curvature. In addition we establish the fundamental theorem for Legendrian submanifolds in Heisenberg groups.
In Chapter 2, we derive the subgradient estimate of the CR heat equation associated with the Witten Laplacian on a closed pseudohermitian (2n+1)-manifold. With its application, we obtain Perelman-type entropy formula for
thse CR heat equation and the CR heat equation associated with the Witten Laplacian.
In Chapter 3, we obtain the representation of PSH(n) which is the group of pseudohermitian transformations on n-dimensional Heisenberg groups. Also we discuss how the matrix Lie group PSH(n) interpret as the set of "frames" on the homogeneous space PSH(n)/U(n). Then for the (left-invariant) Maurer-Cartan form, we immediately get the moving frame formula.
In Chapter 4, we use Élie Cartan’s method of moving frames, the theory of Lie groups to obtain the fundamental theorem for the Legendrian submanifolds in Hesenberg groups. Let Σ be a n-dimensional oriented surface and f:Σ-->H^n be an embedding as a Legendrian submanifold in H^n. For every totally real point p in Σ, we compute the Darboux derivative of the lifting of f to get the integrability conditions for Σ. Then we show that for any Riemannian manifold which satisfies the integrability conditions can be
locally embedded into H^n as a Legendrian submanifold.
關鍵字(中) ★ Perelman的 W-熵公式
★ Witten拉普拉斯
★ Bakry-Emery瑞奇曲率
★ Legendrian子流形
★ 擬埃爾米特流形
關鍵字(英) ★ Perelman’s W-Entropy formula
★ the Witten Laplacian
★ the Bakry-Emery Ricci curvature
★ Legendrian submanifolds
★ pseudohermitian manifolds
論文目次 1 Preliminary 1
1.1 Pseudohermitian Manifolds 1
1.2 Calculus on a Lie Group 4
2 Perelman’s Entropy Formula on Pseudohermitian Manifolds 5
2.1 Perelman-Type Entropy Formula for the CR Heat Equation Associated with the Witten Laplacian 6
2.2 The CR Bochner Formulae 12
2.3 The Li-Yau Subgradient Estimate 17
2.4 Perelman-Type Entropy Formula 23
3 The Group of Pseudohermitian Transformations on Heisenberg Groups 32
3.1 The Pseudohermitian Transformations on Heisenberg Groups 32
3.2 Representation of PSH(n) 38
3.3 The Oriented Frames on Heisenberg Groups 39
3.4 Moving Frame Formula 40
4 the Fundamental Theorem for Legendrian Submanifolds in Heisenberg Groups 41
4.1 Introduction 41
4.2 The Integrability Conditions for Legendrian Submanifolds 43
4.3 The Proof of Fundamental Theorem for Legendrian Submanifolds 47
Bibliography 48
參考文獻 [C] C. Chevalley; Theory of Lie Groups (Princeton University
Press, Princeton, 1946).
[CC] O. Calin and D.-C. Chang; Sub-Riemannian Geometry: General
Theory and Examples (Cambridge ; New York : Cambridge University Press,
2009).
[CC1] 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.
[CC2] S.-C. Chang and H.-L. Chiu; On the CR Analogue of Obata’s Theorem in a Pseudohermitian 3-Manifold, Math. Ann. vol 345, no. 1 (2009), 33-51.
[CCG] O. Calin, D.C. Chang and P. Greiner; Geometric Analysis on the Heisenberg Group and Its Generalizations, American Mathematical Society, 2007.
[CCL] S.S. Cheng, W.H. Cheng and K.S. Lam; Lectures on
Differential Geometry, Singapore; River Edge, N.J.: World Scientific, 1999.
[Ch] B. Chow; The Yamabe flow on locally conformally flat
manifolds with positive Ricci curvature, Comm. Pure and Appl. Math.,XLV(1992),1003-1014.
[CHMY] J.-H. Cheng, J.-F. Hwang, A. Malchiodi and P. Yang; A
Codazzi-Like Equation and the Singular Set for C^1 Smooth surfaces In the Heisenberg Group. J. reine angew. Math. 671 (2012), 131-198.
[Cho] W.-L. Chow; Uber System Von Lineaaren Partiellen
Differentialgleichungen erster Orduung,. Math. Ann. 117 (1939), 98-105.
[ChL] H.-L. Chiu, S.-H. Lai; The Fundamental Theorems for
Curves and Surfaces in 3D Heisenberg Group, preprint.
[CL] W. S. Cohn and G. Lu; Best Constants for Moser-Trudinger Inequalities on the Heisenberg Group, Indiana Univ. Math. J. 50 (2001), 1567-1591.
[CKL1] S.-C. Chang, T.-J. Kuo and S.-H. Lai; Li-Yau Gradient
Estimate and Entropy Formulae for the CR heat equation in a Closed Pseudohermitian 3-manifold, J. Differential Geometry, 89 (2011), 185-216.
[CKL2] S.-C. Chang, T.-J. Kuo and S.-H. Lai; Li-Yau Gradient
Estimate and Entropy Formulae for the Witten Laplacian via the Bakry-Emery Pseudohermitian Ricci Curvature, preprint.
[CSW] J.-T. Chen, T. Saotome and C.-T. Wu; The CR Almost Schur Lemma and Lee Conjecture, preprint.
[CTW] S.-C. Chang, Jingzhu Tie and C.-T. Wu, Subgradient
Estimate and Liouville-type Theorems for the CR Heat Equation on Heisenberg groups H^n, Asian J. Math., Vol. 14, No. 1 (2010), 041--072.
[CW] S.-C. Chang and C.-T. Wu; The Entropy Formulas for the CR Heat Equation and its Applications in a Pseudohermitian (2n+1)-Manifold, Pacific Journal of Mathematics, Vol. 246 (2010), 1--29 .
[CY] 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.
[FH] 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.
[G] P. Griffiths; On Cartan’s Method of Lie Groups and Moving frames as applied to Uniqueness and Existence questions in Differential Geometry, Duke Math. J. 41 (1974), 775-814.
[GG] A. R. Gover and C. R. Graham; CR Invariant Powers of the Sub-Laplacian, J. Reine Angew. Math. 583 (2005), 1-27.
[GL] C. R. Graham and J. M. Lee; Smooth Solutions of Degenerate Laplacians on Strictly Pseudoconvex Domains, Duke Math. J., 57 (1988),
697-720.
[Gr] A. Greenleaf; The first eigenvalue of a Sublaplacian on a Pseudohermitian manifold. Comm. Part. Diff. Equ. 10(2) (1985), no.3 191--217.
[H1] R. S. Hamilton; Three-Manifolds with Positive Ricci
Curvature, J. Diff. Geom. 17 (1982), 255-306.
[H2] R. S. Hamilton; Non-singular Solutions of the Ricci Flow on Three-manifolds, Comm. Anal. Geom. 7 (1999), no. 4, 695-729.
[H3] R. S. Hamilton; The Formation of Singularities in the Ricci Flow. Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 7-136, Internat. Press, Cambridge, MA, 1995.
[H4] R. S. Hamilton; Four-manifolds with Positive Isotropic
Curvature, Comm. Anal. Geom. 5 (1997), no. 1, 1-92.
[Hi] K. Hirachi; Scalar Pseudo-hermitian Invariants and the Szeg"{o} Kernel on 3-dimensional $CR$ Manifolds, Lecture Notes in Pure and ppl. Math. 143, pp. 67-76, Dekker, 1992.
[Ho] L. Hormander; Hypoelliptic Second Order Differential
Equations, Acta Math. 119 (1967), 147-171.
[IL] T.A. Ivey and J.M. Landsberg; Cartan for
Beginners:Differential Geometry via Moving Frames and Exterior Differential Systems. Graduate Studies, in Math. v.61 (2003).
[JS] D. Jerison and A. S’{a}nchez-Calle; Estimates for the Heat Kernel for the Sum of Squares of Vector Fields, Indiana J. Math. 35 (1986), 835-854.
[Le1] J.M. Lee; The Fefferman metric and pseudohermitian
invariants. Trans. Am. Math.Soc. 296 (1986), 411-429.
[Le2] J.M. Lee; Pseudo-Einstein structures on CR manifolds.
Am. J. Math. 110 (1988), 157-178.
[Li1] X. -D. Li; Liouville theorems for symmetric diffusion
operators on complete Riemannian manifolds, J.Math.Pures Appl. 84 (2005), 1295-1361.
[Li2] X. -D. Li; Perelman’s Entropy Formula for the Witten
Laplacian on Riemannian Manifolds via Bakry-Emery Ricci Curvature, Math. Ann. , 353 (2012), No. 2, 403-437.
[LY] P. Li and S.-T. Yau; On the Parabolic Kernel of the Schrӧdinger Operator, Acta Math. 156 (1985), 153-201.
[LY1] P. Li and S.-T. Yau; Estimates of Eigenvalues of a
Compact Riemannian Manifold, AMS Proc. Symp. in Pure Math. 36 (1980),
205-239.
[Na] J. Nash; Continuity of Solutions of Parabolic and Elliptic Equations, Am. J. Math., 80 (1958), 931-954.
[Ni] Lei Ni; The entropy formula for the linear heat equation, J. Geom. Analysis 14 (2004), No. 1, 85-98.
[P] S. Paneitz; A Quartic Conformally Covariant Differential
Operator for Arbitrary Pseudo-Riemannian Manifolds, preprint, 1983.
[Pe1] G. Perelman; The Entropy Formula for the Ricci Flow and its Geometric Applications. ArXiv: Math. DG/0211159.
[Pe2] G. Perelman; The Ricci Flow with Surgery on
Three-manifolds, ArXiv:Math.DG/0303109.
[Pe3] G. Perelman; Finite Extinction Time for the Solutions to the Ricci Flow on Certain Three-manifolds, ArXiv: Math.DG/0307245.
[PT] R. Palais and C.-L. Terng; Critical point theory and submanifold geometry, Berlin ; New York : Springer-Verlag, 1988.
[R] Q.H.Ruan; Bakry-Emery Curvature Operator and Ricci Flow,
Potential Analysis, 25(2006), No.4,399-406.
[S] R.W. Sharp; extbf{Differential Geometry }Cartan’s
Generalization of Klein’s Erlangen Program. Graduate Texts, in Math. v.166 (1997).
[SC] A. Sảnchez-Calle; Fundamental Solutions and Geometry of
the Sum of Squares of Vector Fields, Invent. Math. 7 (1984), 143-160.
[SY] R. Schoen and S.-T. Yau; Lectures on Differential Geometry, International Press, 1994.
[W] E. Witten; Supersymmetry and Morse theory. J. Differ. Geom.17 (1982), 661--692.
[We] S.M. Webster; Pseudo-Hermitian structures on a real
Hypersurface. J. Diff. Geom. 13 (1978), 25-41.
[Wu] L.-M. Wu; Uniqueness of Nelson’s Diffusions, Probab. Theory and Related Fields, 114 (1999), 549--585.
[Y1] S. -T. Yau; Harmonic functions on complete Riemannian
manifolds, Comm. Pure Appl. Math. 28 (1975), 201-228.
[Y2] S. -T. Yau; Seminar on Differential Geometry, edited,
Annals of Math. Studies 102, Princeton, New Jersey, 1982.
指導教授 張樹城、邱鴻麟
(Shu-Cheng Chang、Hung-Lin Chiu)
審核日期 2013-7-22
推文 facebook   plurk   twitter   funp   google   live   udn   HD   myshare   reddit   netvibes   friend   youpush   delicious   baidu   
網路書籤 Google bookmarks   del.icio.us   hemidemi   myshare   

若有論文相關問題,請聯絡國立中央大學圖書館推廣服務組 TEL:(03)422-7151轉57407,或E-mail聯絡  - 隱私權政策聲明