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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 鄭日新 | zh_TW |
| dc.contributor.advisor | Jih-Hsin Cheng | en |
| dc.contributor.author | 高尉庭 | zh_TW |
| dc.contributor.author | Wei-Ting Kao | en |
| dc.date.accessioned | 2025-08-14T16:04:34Z | - |
| dc.date.available | 2025-08-15 | - |
| dc.date.copyright | 2025-08-14 | - |
| dc.date.issued | 2025 | - |
| dc.date.submitted | 2025-08-01 | - |
| dc.identifier.citation | [1] C. Afeltra. A compactness result for the CR yamabe problem in three dimensions. preprint, arXiv:2401.00906, 2023.
[2] L. Andersson, P. T. Chruściel, and H. Friedrich. On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein’s field equations. Comm. Math. Phys., 149(3):587–612, 1992. [3] J.-M. Bony. Principe du maximum, inégalité de harnack et unicité du probleme de cauchy pour les opérateurs elliptiques dégénérés. In Annales de l’institut Fourier, volume 19, pages 277–304, 1969. [4] J.-H. Cheng. Some applications of Cartan’s theory on three-dimensional Cauchy-Riemann geometry. Math. Z., 218(4):527–548, 1995. [5] J.-H. Cheng, H.-L. Chiu, and P. Yang. Uniformization of spherical cr manifolds. Adv. Math., 255:182–216, 2014. [6] J.-H. Cheng, A. Malchiodi, and P. Yang. A positive mass theorem in three dimensional Cauchy-Riemann geometry. Adv. Math., 308:276–347, 2017. [7] J.-H. Cheng, P. Yang, and Y. Zhang. Invariant surface area functionals and singular Yamabe problem in 3-dimensional CR geometry. Adv. Math., 335:405–465, 2018. [8] S. S. Chern and J. K. Moser. Real hypersurfaces in complex manifolds. Acta Math., 133:219–271, 1974. [9] W.-L. Chow. Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math. Ann., 117:98–105, 1939. [10] C. Fefferman and E. M. Stein. Hp spaces of several variables. Acta Math., 129(34):137–193, 1972. [11] D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order, volume 224. Springer-Verlag, Berlin, 1977. [12] C. R. Graham. Volume renormalization for singular Yamabe metrics. Proc. Amer. Math. Soc., 145(4):1781–1792, 2017. [13] C. R. Graham and J. M. Lee. Einstein metrics with prescribed conformal infinity on the ball. Adv. Math., 87(2):186–225, 1991. [14] L. Hörmander. Hypoelliptic second order differential equations. Acta Math, 119:147–171, 1967. [15] D. Jerison and J. M. Lee. The Yamabe problem on CR manifolds. J. Differential Geom., 25(2):167–197, 1987. [16] D. Jerison and J. M. Lee. Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem. J. Amer. Math. Soc., 1(1):1–13, 1988. [17] J. J. Kohn. Pseudo-differential operators and hypoellipticity. Proc. Sympos. Pure Math., 13:61–69, 1973. [18] J. M. Lee. The Fefferman metric and pseudo-Hermitian invariants. Trans. Amer. Math. Soc., 296(1):411–429, 1986. [19] A. Nagel, E. M. Stein, and S. Wainger. Balls and metrics defined by vector fields. I. Basic properties. Acta Math., 155(1-2):103–147, 1985. [20] M. Rumin. Formes différentielles sur les variétés de contact. J. Differential Geom., 39(2):281–330, 1994. [21] A. Sánchez-Calle. Fundamental solutions and geometry of the sum of squares of vector fields. Invent Math, 78(1):143–160, 1984. [22] C. J. Xu. Subelliptic variational problems. Bull. Soc. Math. France, 118(2):147–169, 1990. [23] C. J. Xu. Regularity for quasilinear second-order subelliptic equations. Comm. Pure Appl. Math., 45(1):77–96, 1992. | - |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/98429 | - |
| dc.description.abstract | 本論文包含一個主要研究主題與兩個次要研究主題。主要研究主題是探討柯西黎曼山邊方程的正則性理論在強擬凸仿埃米爾特流型上並且具有光滑且非奇異的邊界。第一個次要主題為在五維柯西黎曼流型中構造超曲面的柯西黎曼面積不變量；第二個次要主題則探討仿埃米爾特質量的性質。
關於主要研究主題，奇異柯西黎曼山邊度量是研究超曲面之柯西離黎曼不變量的重要工具。我們證明，在具有光滑且非奇異邊界的強擬凸仿埃米爾特流型上，奇異雅馬貝方程存在唯一解，並分析給出其邊界漸近行為，同時將 柯西黎曼面積不變量E2推廣至高維情況。 在五維的柯西黎曼面積不變量方面，我運用柯西黎曼流型的卡當幾何結構，為嵌入於五維 CR 流形中的超曲面建構新的不變量。 最後，在仿埃米爾特質量的研究中，我在柯西黎曼史瓦西上構造出具線性奇異性的切向柯恩拉普拉斯算子的顯式解。 | zh_TW |
| dc.description.abstract | This dissertation consists of one primary topic and two subsidiary topics. The main focus is on the regularity theory for the CR singular Yamabe equation on strongly pseudoconvex (spc) pseudohermitian manifolds with smooth, non-singular boundaries. One subsidiary topic involves the construction of CR area invariants for hypersurfaces in 5-dimensional CR manifolds. The other addresses certain properties of the p-mass.
For the primary topic, the singular Yamabe metric serves as a fundamental tool for studying CR invariants of hypersurfaces. We establish the existence, uniqueness, and boundary asymptotic behavior of singular Yamabe solutions on spc CR manifolds with smooth, non-singular boundaries and provide a generalization of the CR invariant energy E2 to higher-dimensional spc CR manifolds. Regarding the CR area invariant in five dimensions, I exploit the Cartan geometric structure of CR manifolds to construct new invariants associated with embedded hypersurfaces in 5-dimensional CR manifolds. Finally, in the context of the p-mass, I present explicit solutions with line singularities for the tangential Kohn Laplacian on the CR Schwarzschild model. | en |
| dc.description.provenance | Submitted by admin ntu (admin@lib.ntu.edu.tw) on 2025-08-14T16:04:33Z No. of bitstreams: 0 | en |
| dc.description.provenance | Made available in DSpace on 2025-08-14T16:04:34Z (GMT). No. of bitstreams: 0 | en |
| dc.description.tableofcontents | Verification Letter from the Oral Examination Committee i
Acknowledgements iii 摘要 v Abstract vii Contents ix Chapter 1 Introduction 1 Chapter 2 Preliminaries 7 2.1 Carnot-Carathéodory Frames Near the Boundary . . . . . . . . . . . 9 2.2 Approximate solutions . . . . . . . . . . . . . . . . . . . . . . . . . 14 Chapter 3 Weighted pseudohermitian Hölder Spaces and Operators on CR Manifolds with Boundary 19 Chapter 4 Some estimate for Subelliptic operators 25 Chapter 5 Existence and Uniqueness of the Singular Yamabe Equation 31 Chapter 6 Behavior of the Solution Near the Boundary 37 Chapter 7 Singular Yamabe Energy for CR manifolds and examples 43 7.1 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 7.2 Singular Yamabe energy . . . . . . . . . . . . . . . . . . . . . . . . 47 Chapter 8 CR Invariants of Hypersurfaces in Cartan Geometry 49 Chapter 9 Some properties for p-mass 55 9.1 The Exact Form on the CR Schwarzschild Model of Dimension 3 . . 55 References 59 Appendix A — Real Version of the Tanaka-Webster Connection 63 | - |
| dc.language.iso | en | - |
| dc.subject | 仿埃米爾特質量 | zh_TW |
| dc.subject | 柯西黎曼山邊方程 | zh_TW |
| dc.subject | 科西黎曼不變量 | zh_TW |
| dc.subject | CR invariant | en |
| dc.subject | The CR Yamabe equation | en |
| dc.subject | pseudohermitian mass | en |
| dc.title | 柯西黎曼山邊問題：奇異柯西黎曼山邊方程的正則性、超曲面的柯西黎曼不變量與仿埃米爾特質量 | zh_TW |
| dc.title | The CR Yamabe Problem: Regularity of the CR Singular Yamabe Equation, CR Invariants of Hypersurfaces, and the p-Mass | en |
| dc.type | Thesis | - |
| dc.date.schoolyear | 113-2 | - |
| dc.description.degree | 博士 | - |
| dc.contributor.coadvisor | 蔡宜洵 | zh_TW |
| dc.contributor.coadvisor | I-Hsun Tsai | en |
| dc.contributor.oralexamcommittee | 蕭欽玉;邱鴻麟;何柏通;李瑩英 ;蔡忠潤 | zh_TW |
| dc.contributor.oralexamcommittee | Chin-Yu Hsiao;Hung-Lin Chiu;Pak-Tung Ho;Yng-Ing Lee;Chung-Jun Tsai | en |
| dc.subject.keyword | 柯西黎曼山邊方程,科西黎曼不變量,仿埃米爾特質量, | zh_TW |
| dc.subject.keyword | The CR Yamabe equation,CR invariant,pseudohermitian mass, | en |
| dc.relation.page | 65 | - |
| dc.identifier.doi | 10.6342/NTU202502851 | - |
| dc.rights.note | 同意授權(限校園內公開) | - |
| dc.date.accepted | 2025-08-05 | - |
| dc.contributor.author-college | 理學院 | - |
| dc.contributor.author-dept | 數學系 | - |
| dc.date.embargo-lift | 2026-09-29 | - |
| 顯示於系所單位： | 數學系 | |
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