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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 郭孝豪(Siao-Hao Guo) | |
| dc.contributor.author | Sheng-Hao Yang | en |
| dc.contributor.author | 楊盛皓 | zh_TW |
| dc.date.accessioned | 2023-03-19T22:49:49Z | - |
| dc.date.copyright | 2022-08-10 | |
| dc.date.issued | 2022 | |
| dc.date.submitted | 2022-08-04 | |
| dc.identifier.citation | [1] Ben Andrews. “Noncollapsing in mean-convex mean curvature flow”. In: Geometry & Topology 16.3 (2012), pp. 1413–1418. doi: 10.2140/gt.2012.16.1413. [2] Yun Gang Chen, Yoshikazu Giga, and Shun’ichi Goto. “Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations”. In: Journal of Differential Geometry 33.3 (1991), pp. 749–786. doi: 10.4310/jdg/1214446564. [3] Otis Chodosh. “Mean curvature flow (Math 258) lecture notes”. In: (). url: https://web.stanford.edu/~ochodosh/MCFnotes.pdf. [4] Otis Chodosh and Felix Schulze. “Uniqueness of asymptotically conical tangent flows”. In: Duke Mathematical Journal 170.16 (2021), pp. 3601–3657. doi: 10.1215/00127094-2020-0098. [5] Julie Clutterbuck, Oliver C. Schnürer, and Felix Schulze. “Stability of translating solutions to mean curvature flow”. In: Calculus of Variations and Partial Differential Equations 29 (2005), pp. 281–293. [6] Tobias H. Colding and William P. Minicozzi. “Generic mean curvature flow I; generic singularities”. In: Annals of Mathematics 175.2 (2012), pp. 755 [7] Tobias Holck Colding, Tom Ilmanen, and William P. Minicozzi. “Rigidity of generic singularities of mean curvature flow”. In: Publications mathématiques de l’IHÉS 121 (2013), pp. 363–382. [8] Tobias Holck Colding and William P. Minicozzi. “Uniqueness of blowups and Łojasiewicz inequalities”. In: Annals of Mathematics 182.1 (2015), pp. 221–285. issn: 0003486X. [9] Klaus Ecker and Gerhard Huisken. “Interior curvature estimates for hypersurfaces of prescribed mean curvature”. In: Annales de l’I.H.P. Analyse non linéaire 6.4 (1989), pp. 251–260. [10] L. C. Evans and J. Spruck. “Motion of level sets by mean curvature. I”. English. In: J. Differ Geom. 33.3 (1991), pp. 635–681. issn: 0022-040X. doi: 10.4310/jdg/1214446559. [11] Lawrence C. Evans. Partial differential equations. Providence, R.I.: American Mathematical Society, 2010. [12] Zhou Gang, Dan Knopf, and Israel Michael Sigal. “Neckpinch dynamics for asymmetric surfaces evolving by mean curvature flow”. In: (2013). arXiv: 1109.0939 [math.DG]. [13] David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order. Reprint of the 1998 edition. Springer-Verlag, 2001. [14] Richard S. Hamilton. “Four-manifolds with positive curvature operator”. In: Journal of Differential Geometry 24.2 (1986), pp. 153–179. doi: 10.4310/jdg/1214440433. [15] Robert Haslhofer and Bruce Kleiner. “Mean curvature flow with surgery”. In: Duke Mathematical Journal 166.9 (2017), pp. 1591–1626. doi: 10.1215/00127094-0000008X. [16] Gerhard Huisken. “Asymptotic behavior for singularities of the mean curvature flow”. In: Journal of Differential Geometry 31.1 (1990), pp. 285–299. doi: 10.4310/jdg/1214444099. [17] Gerhard Huisken. “Flow by mean curvature of convex surfaces into spheres”. In: Journal of Differential Geometry 20.1 (1984), pp. 237–266. doi: 10.4310/jdg/1214438998. [18] Gerhard Huisken and Carlo Sinestrari. “Mean Curvature Flow Singularities for Mean Convex Surfaces”. In: Calculus of Variations and Partial Differential Equations 8 (1999), pp. 1–14. [19] James Isenberg and Haotian Wu. “Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up”. In: (2017). arXiv: 1603.01664 [math.DG] [20] Olga Ladyzhenskaya, V. A. Solonnikov, and Ural’ceva N. N. Linear and quasilinear equations of parabolic type. American Mathematical Society, 1988. [21] Joel Langer. “A Compactness Theorem for Surfaces with Lp-Bounded Second Fundamental Form.” In: Mathematische Annalen 270 (1985), pp. 223–234. [22] Angenent S.B. and Velázquenz J.J. “Degenerate neckpinches in mean curvature flow.” In: Journal für die reine und angewandte Mathematik 482 (1997), pp. 15–66 [23] Leon Simon. “Asymptotics for a Class of Non-Linear Evolution Equations, with Applications to Geometric Problems”. In: Annals of Mathematics 118.3 (1983), pp. 525–571. issn: 0003486X. [24] Juan J. L. Velazquez. “Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow”. In: Annali della Scuola Normale Superiore di Pisa - Classe di Scienze Ser. 4, 21.4 (1994), pp. 595–628. [25] Lu Wang. “Asymptotic structure of self-shrinkers”. In: (2016). arXiv: 1610.04904 [math.DG]. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/85200 | - |
| dc.description.abstract | 這篇文章回顧了圓柱型奇異點的一些結果，特別是退化情況。首先是 AngenentVelázquenz 的特殊解，以及 Colding-Minicozzi 關於收斂到圓柱的結果 | zh_TW |
| dc.description.abstract | Several past results on the cylindrical singularity and in particular its degenerated case where there is a tip region is reviewed. First is Angenent-Velázquenz example, which follows by Colding-Minicozzi result on the convergence to the unit multiplicity cylinder. | en |
| dc.description.provenance | Made available in DSpace on 2023-03-19T22:49:49Z (GMT). No. of bitstreams: 1 U0001-0208202218542200.pdf: 1333052 bytes, checksum: 5f6a5386890f491e411eb4fdd2bcdb51 (MD5) Previous issue date: 2022 | en |
| dc.description.tableofcontents | 口試委員審定書 i Acknowledgements ii 中文摘要 iii Abstract iv 1 Introduction 1 2 1 Shrinking Cylinder with Type-II Blowup Caps 3 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Existence up to blowup time T . . . . . . . . . . . . . . . . . . . . . . 5 2.2.1 Curvature Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.2 Degree Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 2.3 The Convergence on the Cylinder . . . . . . . . . . . . . . . . . . . . . 8 2.4 The Convergence of Type-II Blowup to Bowl Solitons . . . . . . . . . . . . 8 2.4.1 Proof of Derivative q(z, s) Convergence . . . . . . . . . . . . . . . . 9 2.4.2 The Bound of q(z, s)/z . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4.3 The Derivative Bounds of q(z, s) . . . . . . . . . . . . . . . . . . . .11 2.4.4 Convergence of q(z, s)/z . . . . . . . . . . . . . . . . . . . . . . . .12 3 The Convergence to Cylinders via Lojasiewicz Inequality 13 3.1 Lojasiewicz Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.1.1 The Unit Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1.2 Rescaled MCF as an Evolution of Graph over a Cylinder . . . . . . . . . 17 3.1.3 Fredholm Alternative for L . . . . . . . . . . . . . . . . . . . . . . .18 3.1.4 Eigenfunctions of L on Cylinders . . . . . . . . . . . . . . . . . . . .19 3.1.5 Gradient Lojasiewicz . . . . . . . . . . . . . . . . . . . . . . . . . .20 3.2 Growth Rate of Cylindrical Scale . . . . . . . . . . . . . . . . . . . . .22 3.2.1 The Geometry of Cylinders . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.2 The Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . .26 3.2.3 White Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4 A Final Thought 28 Reference 28 | |
| dc.language.iso | en | |
| dc.subject | 退化 Neckpinch | zh_TW |
| dc.subject | 圓柱型奇異點 | zh_TW |
| dc.subject | 平均曲率流 | zh_TW |
| dc.subject | 長時間穩定性 | zh_TW |
| dc.subject | Lojasiewicz 不等式 | zh_TW |
| dc.subject | Lojasiewicz Inequality | en |
| dc.subject | Mean Curvature Flow | en |
| dc.subject | Cylindrical Singularity | en |
| dc.subject | Degenerate Neckpinch | en |
| dc.subject | Long-time Stability | en |
| dc.title | 平均曲率流的圓柱型奇異點的回顧 | zh_TW |
| dc.title | A Review on Cylindrical Singularity in Mean Curvature Flow | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 110-2 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 崔茂培(Mao-Pei Tsui),蔡忠潤 (Chung-Jun Tsai) | |
| dc.subject.keyword | 平均曲率流,圓柱型奇異點,退化 Neckpinch,長時間穩定性,Lojasiewicz 不等式, | zh_TW |
| dc.subject.keyword | Mean Curvature Flow,Cylindrical Singularity,Degenerate Neckpinch,Long-time Stability,Lojasiewicz Inequality, | en |
| dc.relation.page | 30 | |
| dc.identifier.doi | 10.6342/NTU202201987 | |
| dc.rights.note | 同意授權(限校園內公開) | |
| dc.date.accepted | 2022-08-04 | |
| dc.contributor.author-college | 理學院 | zh_TW |
| dc.contributor.author-dept | 數學研究所 | zh_TW |
| dc.date.embargo-lift | 2022-08-10 | - |
| 顯示於系所單位： | 數學系 | |
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