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研究生： 賴冠合
Lai, Guan-Ho
論文名稱： Eigenvalue Estimates on Minimal Submanifolds
指導教授： 宋瓊珠
Sung, Chiung-Jue
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2010
畢業學年度： 98
語文別： 英文
論文頁數： 28
中文關鍵詞： 特徵值 、極小子流形 、拉普拉斯–貝爾特拉米算子 、保角變換 、完備黎曼流形
外文關鍵詞： Eigenvalue, Minimal Submanifold, Laplace–Beltrami operator, Conformal Map, Complete Riemannian Manifold
相關次數： 點閱：2  下載：0 
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    • In this eport, we introduce several theorems about eigenvalues of the
    Laplace-Beltrami operator on submanifolds in Euclidean spaces, especially indide a
    sphere. We will give some sufficient and necessary conditions for a submanifold being minimal.

    1. Introduction 2. The Takahashi Theorem 3. The Theorem of Yang-Yau 3.1. Main Theorems 9 3.2. Proof of Proposition 3.3 3.3. Proof of Theorem 3.1 3.4. Proof of Theorem 3.2 4. The Theorem of Li-Yau 4.1. Main Theorem 4.2. Application References

    [1] R. Adams, Sobolev Spaces, Academic Press, Inc., 2003.

    [2] I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, Inc., 1984, pp. 309–312.

    [3] I. Chavel, Riemannian Geometry, A Modern Introduction, Cambridge University Press, 2006.

    [4] M. do Carmo, Riemannian Geometry, translated by F. Flaherty, Birh‥auser Boston, 1992.

    [5] J. Jost, Compact Riemann Surfaces, Springer, 2006.

    [6] J. Jost, Riemannian Geometry and Geometric Analysis, Springer, 2008.

    [7] R. C. McOwen, Partial Differential Equations, Pearson Education, Inc., 2003.

    [8] T. Takahashi, Minimal Immersion of Riemannian Manifolds, J. Math. Soc. Japan 8 (1966), no. 4.

    [9] P. Li and S. T Yau, A New Conformal Invariant and Its Applications to the Willmore Conjecture
    and the First Eigenvalue of Compact Surfaces, Invent. Math. 69 (1982), 269–291

    [10] R. Schoen and S. T. Yau, Lectures on Differential Geometry, International Press of Boston, 1994

    [11] P. Yang and S. T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal
    submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 7 (1980), no. 1, 55–63

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