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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 崔茂培(MAO-PEI TSUI) | |
| dc.contributor.author | KUAN-HUI LEE | en |
| dc.contributor.author | 李冠輝 | zh_TW |
| dc.date.accessioned | 2021-06-17T07:06:30Z | - |
| dc.date.available | 2019-09-01 | |
| dc.date.copyright | 2019-08-05 | |
| dc.date.issued | 2019 | |
| dc.date.submitted | 2019-07-25 | |
| dc.identifier.citation | [1] M. F. Atiyah and N. Hitchin. Low-energy scattering of non-abelian magnetic monopoles [and discussion]. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences,315(1533):459-469, 1985.
[2] M. F. Atiyah and N. Hitchin. The Geometry and Dynamics of Magnetic Monopoles. Princeton University Press, 1988. [3] B. Chow and D. Knopf. Ricci Flow: an introduction. AMS, 2004. [4] D. M. DeTurck. Deforming metrics in the direction of their ricci tensors. J. Differential Geom., 18(1):157-162, 1983. [5] M. do Carmo. Riemannian Geometry. Springer, 1992. [6] S. K. Donaldson. Moment maps and diffeomorphisms. Asian J. Math., 3(1):1-15, 1999. [7] T. Eguchi and A. J. Hanson. Self-dual solutions to euclidean gravity. Annals of Physics, 120:82-106, 1979. [8] G. W. Gibbons and C. N. Pope. The positive action conjecture and asymptotically euclidean metrics in quantum gravity. Comm. Math. Phys, 66(3):267-290, 1979. [9] D. Gilbarg and N. S.Trudinger. Elliptic Partial Di erential Equations of Second Order. Springer, 1998. [10] N. Hitchin. Hyper-Kähler manifolds. Ast erisque, (206):Exp. No. 748, 3, 137-166, 1992. Seminaire Bourbaki,Vol. 1991/92. [11] N. J. Hitchin. The moduli space of special lagrangian submanifolds. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Ser. 4, 25(3-4):503-515, 1997. [12] G. Huisken. Flow by mean curvature of convex surfaces into spheres. In Miniconference on nonlinear analysis (Canberra, 1984), volume 8 of Proc. Centre Math. Anal. Austral. Nat. Univ., pages 107-112. Austral. Nat. Univ., Canberra, 1984. [13] G. Huisken. Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent. Math., 84(3):463-480, 1986. [14] D. Joyce. Compact manifolds with special holonomy. Oxford University Press, 2000. [15] P. Li. Geometric Analysis. Cambridge Studies in advanced Mathematics. Cambridge University Press,2012. [16] K. Smoczyk. A canonical way to deform a Lagrangian submanifold. arXiv:dg-ga/9605005, 1996. [17] J. Song and B. Weinkove. On Donaldson's flow of surfaces in a hyperkähler four-manifold. Math. Z., 256(4):769-787, 2007. [18] M. B. Stenzel. Ricci-flat metrics on the complexification of a compact rank one symmetric space.manuscripta mathematica, 80:151{163, 1993. [19] C.-J. Tsai and M.-T. Wang. A strong stability condition on minimal submanifolds and its implications.arXiv:1710.00433, 2017. [20] C.-J. Tsai and M.-T. Wang. Global uniqueness of the minimal sphere in the Atiyah-Hitchin manifold.arXiv:1804.08201, 2018. [21] C.-J. Tsai and M.-T. Wang. Mean curvature flows in manifolds of special holonomy. J. Differential Geom.,108(3):531-569, 2018. [22] M.-T. Wang.Mean curvature flow of surfaces in Einstein four-manifolds. J. Di erential Geom., 57(2):301-338, 2001. [23] M.-T.Wang. Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension.Invent. Math., 148(3):525-543, 2002. [24] M.-T. Wang. Subsets of Grassmannians preserved by mean curvature flows. Comm. Anal. Geom.,13(5):981-998, 2005.64 | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/72798 | - |
| dc.description.abstract | 近幾年王慕道教授和蔡忠潤教授發表了一系列關於平均曲率流的論文[19][20][21]，他們證明當一個子流型若C1靠近一個嚴格穩定的極小子流型，那麼平均曲率流就會存在並收斂到此嚴格穩定極小子流型，另一方面，在1999 年Donaldson [6]構造出一系列的幾何流，Song和Weinkove [17]探討了四維超凱勒流型的情況並得到了一些結果，他們發現說在這個情況下唐納森流和平均曲率流是蠻相似的，所以在此碩士論文，我們將完整介紹唐納森流並證明一個類似於王教授和蔡教授的結果到唐納森流上。 | zh_TW |
| dc.description.abstract | In recent year, Wang and Tsai [19][20][21] proposed a series of paper about the stability of mean curvature flow about strongly stable submanifolds. They show that if a submanifold is C1 close to the strongly stable submanifold then the mean curvature flow exists for all time and converges smoothly. On the other hand, Donaldson [6] used moment map and diffeomorphism to construct lots of geometric evolution flows. In particular, the hyperkähler four manifold case was explicitly discussed by Song and Weinkove [17]. They found that Donaldson’s flow is similar to the mean curvature flow in this case. In this thesis, we discuss the Donaldson’s flow in detail and prove a result similar to the Wang and Tsai’s result. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-17T07:06:30Z (GMT). No. of bitstreams: 1 ntu-108-R06221002-1.pdf: 990997 bytes, checksum: f365635295c80d2c6a3c01e11616f3c1 (MD5) Previous issue date: 2019 | en |
| dc.description.tableofcontents | 1 Introduction 2
2 Hyperkähler Four manifolds 4 2.1 Holonomy 4 2.2 Hyperkähler manifolds 4 2.3 Lagrangian submanifold 7 3 Bianchi IX metric and examples 9 3.1 Bianchi IX metric 9 3.2 Eguchi-Hanson space 11 3.2.1 Hyperkähler structure of E-H space 12 3.2.2 Properties of zero section 13 3.3 Atiyah-Hitchin manifolds 14 3.3.1 Geomtry near zero section 14 3.3.2 Curvature 15 3.3.3 Properties of zero section 16 4 Hyperkähler Flow of Donaldson 19 4.1 Introduction and short time existence 19 4.2 Variational viewpoint of H-flow 21 4.3 Evolution formula 22 4.3.1 Basic properties 23 4.3.2 Evolution of second fundamental form 24 4.3.3 Evolution of form 26 4.4 Special hyperkähler Flow of Donaldson 28 4.4.1 Existence 29 4.4.2 Evolution of special hyperkähler flow 32 4.5 Long time existence 35 5 The stability of special H-flow in strongly stable surface 37 5.1 Preliminary 37 5.2 Previous Estimate 39 5.3 C0 estimate 44 5.4 C1 estimate 45 5.5 Long time existence 47 5.6 Proof of main result 48 5.7 Conclusion and Mean Curvature Flow case 50 A Left-invariant one-form in SU(2) 51 B Moving frame method 52 C Submanifold geometry 54 D Second variational formula and strongly stable surface 58 E Maximum Principle 61 References 63 | |
| dc.language.iso | en | |
| dc.subject | 平均曲率流 | zh_TW |
| dc.subject | 超凱勒流型 | zh_TW |
| dc.subject | 唐納森流 | zh_TW |
| dc.subject | Hyperkahler Manifolds | en |
| dc.subject | Mean curvature Flow | en |
| dc.title | 唐納森流和平均曲率流在四維超凱勒流型的穩定性 | zh_TW |
| dc.title | The Stability of Donaldson’s Flow and Mean curvature Flow in Hyperkähler Four Manifolds | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 107-2 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 蔡忠潤,李瑩英,王慕道 | |
| dc.subject.keyword | 超凱勒流型,平均曲率流,唐納森流, | zh_TW |
| dc.subject.keyword | Hyperkahler Manifolds,Mean curvature Flow, | en |
| dc.relation.page | 64 | |
| dc.identifier.doi | 10.6342/NTU201901895 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2019-07-25 | |
| 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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