臺灣博碩士論文加值系統

本論文分為兩部份：第一部份是研究緊緻流形間的均曲率流，論文中將王慕道於2002年[19]與崔茂培、王慕道於2004年[17]這兩篇關於高餘維的均曲率流有長時間存在性與收斂性做進一步地推廣，其特色是流形的截面曲率不限定為常數，以及放寬幾何量*Ω的下界。文章後面也給予兩個關於均曲率流的應用。
第二部份是探討施瓦西(Schwarzschild)時空上的球對稱類空常均曲率超曲面，從分析施瓦西時空的內部與外部的球對稱類空常均曲率超曲面的漸近行為，藉由Kruskal擴張，可以將外部與內部的曲面適當地相接，進而得到整體有定義的曲面。文章的最後一節，我們重新以較清楚的方式討論某一類型的常均曲率層，此常均曲率層的研究曾經由Edward Malec與Niall O Murchadha在2003年的文章中[9]討論過。

The thesis consists of two parts. First part is “Mean curvature flow of the graphs of maps between compact manifolds.” We make several improvements on the results of M.-T. Wang in [19] and his joint paper with M.-P. Tsui [17] concerning the long time existence and convergence for solutions of mean curvature flow in higher co-dimension. Both the curvature condition and lower bound of $*Omega$ are weakened. New applications are also obtained.
Second part is “Spherically symmetric spacelike hypersurfaces with constant mean curvature in Schwarzschild spacetimes.” We analyze all spherically symmetric spacelike constant mean curvature hypersurfaces in Schwarzschild exterior and in Schwarzschild interior. They can be joined by choosing suitable parameters through the Kruskal extension, which is the maximal extension of Schwarzschild metric. We also give another argument for some constant mean curvature foliation in Schwarzschild spacetime, which was ever discussed by Edward Malec and Niall O Murchadha in [9].

Table of Contents
Abstract iii
Table of Contents iv
1 Mean Curvature Flow of the Graphs of Maps Between Compact Manifolds 1
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 The area-decreasing case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Bibliography 31

2 Spherically symmetric spacelike hypersurfaces with constant mean curvature in Schwarzschild spacetimes 33
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 The Kruskal Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1 The Kruskal Extension . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 Level sets r = constant and t = constant in Kruskal extension . . . . 41
2.3 Level sets X = constant and T = constant in Schwarzschild spacetimes 43
2.4 Null geodesics between Schwarzschild spacetime and Kruskal extension 45
3 Spherically symmetric spacelike hypersurfaces with constant mean curvature in Schwarzschild spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1 Spacelike hypersurfaces in Schwarzschild spacetimes . . . . . . . . . . 45
3.2 Spherically symmetric spacelike constant mean curvature hypersurfaces in Schwarzschild exterior r > 2M . . . . . . . . . . . . . . . . . 46
3.3 Asymptotic behavior of spherically symmetric spacelike constant mean curvature hypersurfaces in Schwarzschild exterior . . . . . . . . . . . 49
3.4 Cylindrical hypersurfaces r = constant in Schwarzschild interior . . . 54
3.5 Noncylindrical hypersurfaces in Schwarzschild interior . . . . . . . . . 55
3.6 Asymptotic behavior of spherically symmetric spacelike constant mean curvature hypersurfaces in Schwarzschild interior . . . . . . . . . . . 63
4 Spherically symmetric spacelike constant mean curvature hypersurfaces in Kruskal extension I’ and II’ . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.1 Spherically symmetric spacelike constant mean curvature hypersurfaces in Kruskal extension I’ . . . . . . . . . . . . . . . . . . . . . . 66
4.2 Spherically symmetric spacelike constant mean curvature hypersurfaces in Kruskal extension II’ . . . . . . . . . . . . . . . . . . . . . . 66
5 Globally defined constant mean curvature hypersurfaces in Schwarzschild spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6 Constant mean curvature foliations in Schwarzschild spacetimes . . . . . . . 75
Bibliography 79

PART I:
[1] Ballmann, Werner; Gromov, Mikhael; Schroeder, Viktor: Manifolds of nonpositive curvature. Progress inMathematics, 61. Birkh¨auser Boston, Inc., Boston, MA, 1985. vi+263 pp. ISBN: 0-8176-3181-X

[2] Borel, Armand: Compact Clifford-Klein forms of symmetric spaces. Topology 2 (1963), 111–122.

[3] Brakke, Kenneth A.: The motion of a surface by its mean curvature. Mathematical Notes, 20. Princeton University Press, Princeton, N.J., 1978. i+252 pp. ISBN: 0-691- 08204-9

[4] Cheeger, Jeff; Ebin, David G.: Comparison theorems in Riemannian geometry. North-Holland Mathematical Library, Vol. 9. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. viii+174 pp.

[5] De Vito, Jason: Curvature of Invariant Metrics on Compact Lie Groups. http://www.math.upenn.edu/~devito/curv.pdf

[6] Greene, Robert E.: Isometric embeddings of Riemannian and pseudo-Riemannian manifolds. Memoirs of the American Mathematical Society, No. 97 American Mathematical Society, Providence, R.I. 1970 iii+63 pp.

[7] Guth, Larry: Homotopically non-trivial maps with small k-dilation. arXiv:0709.1241v1

[8] Helgason, Sigurdur: Differential geometry, Lie groups, and symmetric spaces. Corrected reprint of the 1978 original. Graduate Studies in Mathematics, 34. American Mathematical Society, Providence, RI, 2001. xxvi+641 pp. ISBN: 0-8218-2848-7

[9] Huisken, Gerhard: Flow by mean curvature of convex surfaces into spheres. J. Differential Geom. 20 (1984), no. 1, 237–266.

[10] Huisken, Gerhard: Asymptotic behavior for singularities of the mean curvature flow. J. Differential Geom. 31 (1990), no. 1, 285–299.

[11] Ilmanen, Tom: Singularity of mean curvature flow of surfaces. preprint, 1997.

[12] Jost, J¨urgen: Nonpositive curvature: geometric and analytic aspects. Lectures in Mathematics ETH Z¨urich. Birkh¨auser Verlag, Basel, 1997. viii+108 pp. ISBN: 3-7643-5736-3

[13] Kobayashi, Shoshichi; Nomizu, Katsumi: Foundations of differential geometry. Vol. II. Reprint of the 1969 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1996. xvi+468 pp. ISBN: 0-471-15732-5

[14] Leon, Steven J.: Linear algebra with applications. 6th ed., Upper Saddle River, N.J.: Prentice Hall, 2002 xv+544 pp. ISBN: 0-13-033781-1

[15] Li, Guanghan; Salavessa, Isabel M.C.: Mean curvature flow of spacelike graphs in Pseudo-Riemannian manifolds. arXiv:0804.0783

[16] Nash, John: The imbedding problem for Riemannian manifolds. Ann. of Math., (2) 63, (1956), 20–63.

[17] Tsui, Mao-Pei; Wang, Mu-Tao: Mean curvature flows and isotopy of maps between spheres. Comm. Pure Appl. Math. 57 (2004), no. 8, 1110–1126.

[18] Wang, Mu-Tao: Mean curvature flow of surfaces in Einstein four-manifolds. J. Differential Geom. 57 (2001), no. 2, 301–338.

[19] Wang, Mu-Tao: Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension. Invent. Math. 148 (2002), no. 3, 525–543.

[20] White, Brian: A local regularity theorem for mean curvature flow. Ann. of Math. (2) 161 (2005), no. 3, 1487–1519.


PART II:
[1] Andersson, Lars; Iriondo, Mirta S.: Existence of constant mean curvature hypersurfaces in asymptotically flat spacetimes. Ann. Global Anal. Geom. 17 (1999), no. 6, 503–538.

[2] Bartnik, Robert: Existence of maximal surfaces in asymptotically flat space-times. Comm. Math. Phys. 94 (1984), 155–175.

[3] Bartnik, Robert; Simon, Leon: Spacelike hypersurfaces with prescribed boundary values and mean curvature. Comm. Math. Phys. 87 (1982), 131–152.

[4] Brill, Dieter R.; Cavallo, John M.; Isenberg, James A.: K-surfaces in the Schwarzschild space-time and the construction of lattice cosmologies. J. Math. Phys. 21 (1980), no. 12, 2789–2796.

[5] Choi, Hyeong In; Treibergs, Andrejs: Gauss maps of spacelike constant mean curvature hypersurfaces of Minkowski space. J. Differential Geom. 32 (1990), no. 3, 775–817.

[6] Cheng, Shiu Yuen; Yau, Shing Tung: Maximal space-like hypersurfaces in Lorentz-Minkowski spaces. Ann. of Math. 104 (1976), 407–419.

[7] Hawking, S. W.; Ellis, G. F. R: The large scale structure of space-time. Cambridge Monographs on Mathematical Physics, No. 1. Cambridge University Press, London-New York, 1973. xi+391 pp.

[8] Kruskal, M. D.: Maximal extension of Schwarzschild metric. Phys. Rev. (2) 119 (1960), 1743–1745.

[9] Malec, Edward; ´O Murchadha, Niall: Constant mean curvature slices in the extended Schwarzschild solution and the collapse of the lapse. Phys. Rev. D. (3) 68 (2003), no. 12, 124019, 16 pp.

[10] Malec, Edward; ´O Murchadha, Niall: General spherically symmetric constant mean curvature foliations of the Schwarzschild solution, Phys. Rev. D. 80 (2009), no. 2, 024017, 8 pp.

[11] Treibergs, Andrejs: Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, Invent. Math. 66 (1982), no. 1, 39–56.

[12] Wald, Robert M.: General relativity, University of Chicago Press, Chicago, IL, 1984. xiii+491 pp. ISBN: 0-226-87032-4; 0-226-87033-2