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研究生:陳威有
研究生(外文):Chen, Wei-Yu
論文名稱:正常嵌入的極小曲面
論文名稱(外文):complete properly embedded minimal surfaces
指導教授:王譪農
指導教授(外文):Wang, Ai Nung
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱:數學研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2001
畢業學年度:89
語文別:英文
論文頁數:16
中文關鍵詞:極小曲面有限拓撲有限全曲率環型終點廣義Nitsche 猜想
外文關鍵詞:minimal surfacesfinite topologyfinite total curvatureannular endgeneralized Nitsche conjecture
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廣義的Nitsche猜想是如下所述:
假設E是一個正常嵌入在半空間H={(x_{1},x_{2},x_{3})\in R^{3};x_{3}>0}且和每一個平面$x_{3}=t,t \geq 0$相交於單封閉曲線的極小環型終點.且E的邊界是H的邊界中的光滑曲線,則E是有限全曲率.
在這論文藉由對環型終點的研究,我們得知廣義的Ni-tsche猜想是對的.由此,我們可知對一個正常嵌入在$R^{3}$且有至少兩個終點的極小曲面,其為有限拓撲若且唯若其為有限全曲率.

By an annular end of a surface we mean it has a representation homeomorphic to a punctured disk. We study the annular ends of a complete properly embedded minimal surface in 3-dimentions Euclidean, and the relation between finite topology and finite total curvature.

1.Introduction................................................2
2.Notations...................................................5
3.Preliminaty.................................................6
4.Annular end in $R^3$........................................8
5.Biblography................................................15

[1]P.Collin, Topologie et courbure des surfaces minimales proprement plong\'{e}es de $R^{3}$, Annals of Mathematics, 145(1997), 1-31.
[2] T.H.Colding and W.P.Minicozzi, Complete properly embedded minimal surfaces in $R^{3}$, Duke Mathematical journal. 107(2) 2001,421-426
[3] R.Osserman, Global properties of minimal surface in $E^{3}$ and $E^{n}$, Ann. of Math. 80(1964),340-364.
[4] W.H.Meeks and H.Rosenberg, The geometry of periodic minimal surfaces, Comment. Math. Helv. 68(1993), 538-578.
[5] D.Hoffman. W.H.Meeks, The asympotic behavior of properly embedded minimal surfaces of finite topology. J. Am. Math. Soc 2(4)(1989),667-681
[6] W.H.Meeks and H.Rosenberg, The geometry and conformal structure of properly embedded minimal surfaces of finite topology in $R^{3}$, Invent. Math. 114(1993), 625-639
[7] L.Jorge and W.H.Meeks, The topology of complete minimal surfaces of finite total gaussian curvature, Topology 22(1983),203-221
[8] T.H.Colding abd W.P.Minicozzi, Convergence and compactness of minimal surfaces without density bounds,2000.
[9] S.Y.Cheng and S.T.Yau, Differential equations on Riemannian manifolds and their geometric application, Comm. Pure Appl. Math. 28 (1975), 333-354
[10] R.Osserman and M.Schiffer, Doubly-connected minimal surfaces, Arch. Rational Mech. Anal. 58(1975), 285-307
[11] R.Gilbarg and N.Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Varlag,Berlin-New York, 1983.

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