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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Constant-mean-curvature surface ： ウィキペディア英語版 Constant-mean-curvature surface In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature.〔Nick Korevaar, Jesse Ratkin, Nat Smale, Andrejs Treibergs, A survey of the classical theory of constant mean curvature surfaces in R3, 2002 ()〕〔 This includes minimal surfaces as a subset, but typically they are treated as special case. Note that these surfaces are generally different from constant Gaussian curvature surfaces, with the important exception of the sphere. ==History== In 1841 Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. These are the plane, cylinder, sphere, the catenoid, the unduloid and nodoid.〔C. Delaunay, Sur la surface de révolution dont la courbure moyenne est constante, J. Math. Pures Appl., 6 (1841), 309–320.〕 In 1853 J. H. Jellet showed that if $S$ is a compact star-shaped surface in $\backslash R^3$ with constant mean curvature, then it is the standard sphere.〔J. H. Jellet, Sur la Surface dont la Courbure Moyenne est Constant, J. Math. Pures Appl., 18 (1853), 163–167〕 Subsequently A. D. Alexandrov proved that a compact embedded surface in $\backslash R^3$ with constant mean curvature $H\; \backslash neq\; 0$ must be a sphere.〔A. D. Alexandrov, Uniqueness theorem for surfaces in the large, V. Vestnik, Leningrad Univ. 13, 19 (1958), 5–8, Amer. Math. Soc. Trans. (Series 2) 21, 412–416.〕 Based on this H. Hopf conjectured in 1956 that any immersed compact orientable constant mean curvature hypersurface in $\backslash R^n$must be a standard embedded $n-1$ sphere. This conjecture was disproven in 1982 by Wu-Yi Hsiang using a counterexample in $\backslash R^4$. In 1984 Henry C. Wente constructed the Wente torus, an immersion into $\backslash R^3$ of a torus with constant mean curvature. 〔.〕 Up until this point it had seemed that CMC surfaces were rare; new techniques produced a plethora of examples.〔Karsten Grosse-Brauckmann, Robert B. Kusner, John M. Sullivan. Coplanar constant mean curvature surfaces. Comm. Anal. Geom. 15:5 (2008) pp. 985–1023. ArXiv math.DG/0509210. ()〕 In particular gluing methods appear to allow combining CMC surfaces fairly arbitrarily.〔N. Kapouleas. Complete constant mean curvature surfaces in Euclidean three space, Ann. of. Math. (2) 131 (1990), 239–330〕〔Rafe Mazzeo, Daniel Pollack, Gluing and Moduli for Noncompact Geometric Problems. 1996 arXiv:dg-ga/9601008 ()〕 Delaunay surfaces can also be combined with immersed "bubbles", retaining their CMC properties.〔I. Sterling and H. C. Wente, Existence and classification of constant mean curvature multibubbletons of finite and infinite type, Indiana Univ. Math. J. 42 (1993), no. 4, 1239–1266.〕 Meeks showed that there are no embedded CMC surfaces with just one end in $\backslash R^3$.〔Meeks W. H., The topology and geometry of embedded surfaces of constant mean curvature, J. Diff. Geom. 27 (1988) 539–552.〕 Korevaar, Kusner and Solomon proved that a complete embedded CMC surface will have ends asymptotic to unduloids.〔Korevaar N., Kusner R., Solomon B., The structure of complete embedded surfaces with constant mean curvature, J. Diff. Geom. 30 (1989) 465–503.〕 Each end carries a $n(2\backslash pi-n)$ "force" along the asymptotic axis of the unduloid (where n is the circumference of the necks), the sum of which must be balanced for the surface to exist. Current work involves classification of families of embedded CMC surfaces in terms of their moduli spaces.〔John M. Sullivan, A Complete Family of CMC Surfaces. In Integrable Systems, Geometry and Visualization, 2005, pp 237–245. ()〕 In particular, for $k\; \backslash geq\; 3$ coplanar ''k''-unduloids of genus 0 satisfy $\backslash sum\_^k\; n\_i\; \backslash leq\; (k-1)\backslash pi$ for odd ''k'', and $\backslash sum\_^k\; n\_i\; \backslash leq\; k\backslash pi$ for even ''k''. At most ''k'' − 2 ends can be cylindrical.〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Constant-mean-curvature surface」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.