緊緻流形間的均曲率流與施瓦西時空上的常均曲率超曲面

Mean Curvature Flow Between Compact Manifolds and Constant Mean Curvature Hypersurfaces in Schwarzschild Spacetimes

李國瑋(Kuo-Wei Lee)

指導教授 ： 李瑩英

國立臺灣大學/理學院/數學研究所/博士(2010年)

https://doi.org/10.6342/NTU.2010.03021

摘要

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。

關鍵字

均曲率流 ； 高餘維 ； 長時間存在性 ； 施瓦西時空 ； Kruskal 擴張 ； 常均曲率超曲面 ； 常均曲率層.

並列摘要

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].

並列關鍵字

mean curvature flow ； higher codimension ； long time existence ； Schwarzschild spacetime ； the Kruskal extension ； constant mean curvature hypersurfaces ； constant mean curvature foliation.

參考文獻

[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.

[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.

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