Costa 极小曲面

Costa 曲面是一个完备极小嵌入曲面，具有有限拓扑结构（即，它没有边界且不自相交）。它的亏格为 1，有三个穿孔（Schwalbe 和 Wagon 1999）。在 Costa (1984) 发现这个曲面之前，唯一已知的中没有自相交的完备极小可嵌入曲面是平面（亏格 0）、悬链面（亏格 0，有两个穿孔）和螺旋面（亏格 0，有两个穿孔），并且人们曾推测这些是唯一的此类曲面。

令人惊讶的是，Costa 曲面属于二面体对称群。

Costa 极小曲面出现在 Osserman (1986；左图) 的封面上，以及 La Gaceta de la Real Sociedad Matemática Española 第 2 卷第 2 期 (1999；右图) 的封面上。

Snow sculpture of the Costa minimal surface

它也被制成雪雕 (Ferguson 等人 1999, Wagon 1999)。

Invisible Handshake sculpture by Helaman Ferguson

2008 年 2 月 20 日，Helaman Ferguson 的大型石雕安装在麦卡莱斯特学院奥林-赖斯科学中心的南侧平台（照片由 Stan Wagon 提供）。

正如 Gray (Ferguson 等人 1996, Gray 1997) 发现的那样，Costa 曲面可以用参数形式显式表示为

		$1/2R{-zeta(u+iv)+piu+(pi^2)/(4e_1)+pi/(2e_1)[zeta(u+iv-1/2)-zeta(u+iv-1/2i)]}$	(1)
		$1/2R{-izeta(u+iv)+piv+(pi^2)/(4e_1)-pi/(2e_1)[izeta(u+iv-1/2)-izeta(u+iv-1/2i)]}$	(2)
			(3)

其中是 Weierstrass zeta 函数，是 Weierstrass 椭圆函数，其中 (OEIS A133747)，不变量对应于半周期 1/2 和，第一个根为

(4)

(OEIS A133748)，其中是 Weierstrass 椭圆函数。

另请参阅

完备极小曲面, 极小曲面, Weierstrass 椭圆函数, Weierstrass Zeta 函数

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参考文献

Borwein, J. 和 Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 86-87, 2003.Costa, A. "Examples of a Complete Minimal Immersion in

of Genus One and Three Embedded Ends." Bil. Soc. Bras. Mat. 15, 47-54, 1984.do Carmo, M. P. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 43, 1986.

Ferguson, H.; Gray, A.; 和 Markvorsen, S. "Costa's Minimal Surface via Mathematica." Mathematica in Educ. Res. 5, 5-10, 1996. http://library.wolfram.com/infocenter/Articles/2736/.Ferguson, H.; Ferguson, C.; Nemeth, T.; Schwalbe, D.; 和 Wagon, S. "Invisible Handshake." Math. Intell. 21, 30-35, 1999.GRAPE. "Costa's Surface (Celsoe Costa)." http://www-sfb256.iam.uni-bonn.de/grape/EXAMPLES/AMANDUS/costa.html.Gray, A. "Costa's Minimal Surface." §32.5 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 747-757, 1997.Hoffman, D. 和 Meeks, W. H. III. "A Complete Embedded Minimal Surfaces in

with Genus One and Three Ends." J. Diff. Geom. 21, 109-127, 1985.Nordstrand, T. "Costa-Hoffman-Meeks Minimal Surface." http://jalape.no/math/costatxt.Osserman, R. A Survey of Minimal Surfaces. New York: Dover, pp. 149-150, 1986.Peterson, I. "Three Bites in a Doughnut: Computer-Generated Pictures Contribute to the Discovery of a New Minimal Surface." Sci. News 127, 161-176, 1985.Peterson, I. "The Song in the Stone: Developing the Art of Telecarving a Minimal Surface." Sci. News 149, 110-111, Feb. 17, 1996.Ramos Batista, V. "The Doubly Periodic Costa Surfaces." Math. Z. 240, 549-577, 2002.Ramos Batista, V. "A Family of Triply Periodic Costa Surfaces." Pacific J. Math. 212, 347-370, 2003.Ramos Batista, V. "Singly Periodic Costa Surfaces." J. London Math. Soc. 72, 478-496, 2005.Schwalbe, D. 和 Wagon, S. "The Costa Surface, in Show and Mathematica." Mathematica in Educ. Res. 8, 56-63, 1999.Sloane, N. J. A. Sequences A133747 and A133748 in "The On-Line Encyclopedia of Integer Sequences."Wagon, S. "Snow Sculpting with Mathematics." Jan 25, 1999. http://stanwagon.com/snow/breck1999.Wagon, S. "Invisible Handshake." http://stanwagon.com/wagon/Misc/invisiblehandshake.html.Wolfram Research, Inc. "3-D Zoetrope at SIGGRAPH 2000." http://www.wolfram.com/news/zoetrope.html.

请引用本文为

Weisstein, Eric W. "Costa 极小曲面。" 来自 MathWorld--Wolfram Web 资源。 https://mathworld.net.cn/CostaMinimalSurface.html