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http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/6789完整後設資料紀錄
| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 李瑩英(Yng-Ing Lee) | |
| dc.contributor.author | Yang-Kai Lue | en |
| dc.contributor.author | 呂楊凱 | zh_TW |
| dc.date.accessioned | 2021-05-17T09:18:10Z | - |
| dc.date.available | 2012-07-27 | |
| dc.date.available | 2021-05-17T09:18:10Z | - |
| dc.date.copyright | 2012-07-27 | |
| dc.date.issued | 2012 | |
| dc.date.submitted | 2012-07-18 | |
| dc.identifier.citation | [1] H. Anciaux, Construction of Lagrangian self-similar solutions to the mean curvature
flow in Cn. Geom. Dedicata 120 (2006), 37-48. [2] S. Angenent, Shrinking doughnuts, Nonlinear diffusion equations and their equi- librium states, Birkha‥user, Boston-Basel-Berlin, 3, 21-38, 1992. [3] S. B. Angenent, D. L. Chopp, and T. Ilmanen. A computed example of nonuniqueness of mean curvature flow in R3. Comm. Partial Differential Equa- tions, 20 (1995), no. 11-12, 1937-1958. [4] U. Abresch and J. Langer, The normalized curve shortening flow and homothetic solutions. J. Differential Geom. 23 (1986), no. 2, 175-196. [5] B. Andrews; H. Li; Y. wei, F-stability for self-shrinking solutions to mean curvature flow, preprint, 2012, http://arxiv.org/abs/1204.5010. [6] T.H. Colding and W.P. Minicozzi, Generic mean curvature flow I; generic singularities, to appear in Ann. Math. [7] G. Huisken, Flow by mean curvature of convex surfaces into spheres. J. Differ- ential Geom. 20 (1984), no. 1, 237–266. [8] G. Huisken, Asymptotic behavior for singulairites of the mean curvature flow. J. Differential Geom. 31 (1990), no. 1, 285–299. 42 [9] G. Huisken, Local and global behaviour of hypersurfaces moving by mean curvature. Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., 54, Part 1, Amer. Math. Soc., Providence, RI, (1993), 175-191. [10] R. Harvey; H.B. Lawson, Calibrated geometries. Acta Math. 148 (1982), 47-157. [11] T. Ilmanen, Singularities of mean curvature flow of surfaces, preprint, 1995, http://www.math.ethz.ch/ ilmanen/papers/pub.html. [12] D. Joyce; Y-I Lee; M-P Tsui, Self-similar solutions and translating solitions for Lagrangian mean curvature flow. J Differential Geom. 84 (2010), no. 1, 127-161. 53C44 (53D12). [13] Stephen Kleene and Niels Martin Moller, self-shrinkers with a rotational symmetry, preprint, http://arxiv.org/abs/1008.1609. [14] Y.-I. Lee and M.-T. Wang, Hamiltonian stationary cones and self-similar solutions in higher dimension, Trans. Amer. Math. Soc. 362 (2010), 1491–1503. [15] K. Smoczyk, A canonical way to deform a Lagrangian submanifold, preprint, http://arxiv.org/pdf/dg-ga/9605005v2.pdf. [16] K. Smoczyk, Self-shrinkers of the mean curvature flow in arbitrary codimension, International Mathematics Research Notices, 48 (2005), 2983-3004. [17] A. Stone, A density function and the structure of singularities of the mean curvature flow. Calc. Var. Partial Differential Equations 2 (1994), no. 4, 443-480. [18] B. White, A local regularity theorem for classical mean curvature flow. Ann. of Math. (2) 161 (2005), no. 3, 1487-1519. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/6789 | - |
| dc.description.abstract | In this thesis, we generalize Colding and
Minicozzi's work on the stability of self-shrinkers in the hypersurface case to higher co-dimensional cases. The 1st and 2nd variation formulae of the $F$-functional are derived and an equivalent condition to the stability in general codimension is found. Using the equivalent condition, we can classify $F$-stable product self-shrinkers and show that the Lagrangian self-shrinkers given by Anciaux are $F$- unstable. | en |
| dc.description.provenance | Made available in DSpace on 2021-05-17T09:18:10Z (GMT). No. of bitstreams: 1 ntu-101-D95221005-1.pdf: 432448 bytes, checksum: d41908a4f05e90526f822e15683f06fb (MD5) Previous issue date: 2012 | en |
| dc.description.tableofcontents | Contents
Acknowledgements i Abstract (in Chinese) ii Abstract (in English) iii Contents iv 1 Introduction 1 2 The 1st and 2nd variation formulae of F 6 2.1 Notation and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 The first variation formula of F . . . . . . . . . . . . . . . . . . . . . 8 2.3 The general second variation formula of F . . . . . . . . . . . . . . . 10 2.4 The second variation at a critical point . . . . . . . . . . . . . . . . . 12 3 An equivalent condition for F-stability 15 3.1 Vector-valued eigenfunctions and eigenvalues of L^⊥ . . . . . . . . . . 15 3.2 An equivalent condition . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 Classification of stable product self-shrinkers 21 4.1 For compact case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 For noncompact case . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5 The unstability of Anciaux’s examples 24 5.1 Anciaux’s examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.2 The unstability for general variations . . . . . . . . . . . . . . . . . . 25 5.3 The unstability for Lagrangian variations . . . . . . . . . . . . . . . . 32 6 Self-similar Lagrangian graph 37 6.1 Expanding Lagrangian graph . . . . . . . . . . . . . . . . . . . . . . 37 6.2 Translating Lagrangian graph . . . . . . . . . . . . . . . . . . . . . . 39 | |
| dc.language.iso | en | |
| dc.title | 均曲率流的拉格拉奇自同構解 | zh_TW |
| dc.title | Lagrangian Self-Similar Solutions for Mean Curvature Flow | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 100-2 | |
| dc.description.degree | 博士 | |
| dc.contributor.oralexamcommittee | 王慕道,鄭日新,蔡東和,張樹城 | |
| dc.subject.keyword | 拉格拉奇,同構解,均曲率, | zh_TW |
| dc.subject.keyword | Self-Similar solution,Lagrangian,mean curvature vector, | en |
| dc.relation.page | 43 | |
| dc.rights.note | 同意授權(全球公開) | |
| dc.date.accepted | 2012-07-18 | |
| 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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