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研究生： 郭庭榕
Ting-Jung Kuo
論文名稱： Willmore泛函之探討
The Willmore Functional
指導教授： 張樹城
Shu-Chen Chang
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： 2006
畢業學年度： 94
語文別： 英文
論文頁數： 38
中文關鍵詞： Willmore泛函 、均曲率向量
外文關鍵詞： Willmore functional, mean curvature vector
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    • 探討Willmore functional在二維曲面以及高維度流形的情況,在這篇論文裡分成兩部分,第一部分主要將重點放在二維度曲面的情況,江探討Willmore functional 的基本性質,以及與拉普拉斯算子的第一特徵值之間的關係,以及與conformal area之間的關係.
    第二部分主要是想將Willmore functional推廣到高維度流形的情況,並計算其Euler-Lagrange Equation並找出其Willmore submanifolds的充分必要條件

    In this thesis, we consider the Willmore functional both on surfaces and general Riemannian manifolds. In surfaces case, we study some basic properties of Willmore functional, for example, the relation between conformal area, and the first eigenvalue..., ect. In general case, we calculate the Euler-Lagrange Equation for Willmore functional.

    Abstract Acknowledgement 1. Introduction 1 Part 1. The Willmore functional for surfaces 1 2. Willmore functional and first eigenvalue of Laplacian 3 3. The existence of surface minimizing the Willmore functional 6 4. Conformal invariant for Willmore functional 8 5. Willmore functional and conformal area 15 Part 2. The Willmore functional for high dimensional submanifold in general 18 Riemannian manifold 6. The generalized Willmore functional 18 7. Basic knowledges for submanifold 21 8. The Euler Lagrange equation for the Willmore functional 24 References 38

    Chen1: Bang-yen Chen, On a variational problem of hypersurfaces, J. London Math. Soc., 6 (1972), 321-325.
    Chen2 : Bang-yen Chen, Some conformal invariants of submanifolds and their applications, Boll. Un. Mat. Ital.,(4) 10 (1974), 380-385.
    Chen3 : Bang-yen Chen, Total Mean Curvature and Submanifolds of Finite Type, World Scientific Pub. Co. Pte. Ltd., 1984.
    Chang1 : Shu-cheng Chang, The Willmore Flow On 2-Spheres.(preprint)
    HL : Z.hu and H.li, Willmore Submanifold in a Riemannian Manifold, Contemporary Geometry and Related Topics, 2004, 251-276
    Sim 1 : L.M. Simon, Existence of Surfaces Minimizing the Willmore Functional, Comm. Anal. Gemo. 1 (1993), 281-326.
    Sim 2 : L.M. Simon, Lectures on Geometric Measure Theory, Proc. Centre for Math.Anal., 3 (1983).
    SY : R. Schoen and S.-T. Yau, Lectures on Differential Geometry, International Press, Boston 1994.
    Wil 1 : T.J. Willmore, Total Curvature in Riemannian Geometry, John Wiley and Sons, 1982, Chichester.
    Wil 2 : T.J. Willmore, Riemannian Geometry, Oxford University Press Inc, New York, 1993.

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