What Are Minimal Surfaces?
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Brief Information on the Surfaces Not Included in the Basic Content of the Encyclopedia
Brief Information on the Surfaces Not Included in the Basic Content of the Encyclopedia Brief information on some classes of the surfaces which cylinders, cones and ortoid ruled surfaces with a constant were not picked out into the special section in the encyclo- distribution parameter possess this property. Other properties pedia is presented at the part “Surfaces”, where rather known of these surfaces are considered as well. groups of the surfaces are given. It is known, that the Plücker conoid carries two-para- At this section, the less known surfaces are noted. For metrical family of ellipses. The straight lines, perpendicular some reason or other, the authors could not look through to the planes of these ellipses and passing through their some primary sources and that is why these surfaces were centers, form the right congruence which is an algebraic not included in the basic contents of the encyclopedia. In the congruence of the4th order of the 2nd class. This congru- basis contents of the book, the authors did not include the ence attracted attention of D. Palman [8] who studied its surfaces that are very interesting with mathematical point of properties. Taking into account, that on the Plücker conoid, view but having pure cognitive interest and imagined with ∞2 of conic cross-sections are disposed, O. Bottema [9] difficultly in real engineering and architectural structures. examined the congruence of the normals to the planes of Non-orientable surfaces may be represented as kinematics these conic cross-sections passed through their centers and surfaces with ruled or curvilinear generatrixes and may be prescribed a number of the properties of a congruence of given on a picture. -
Embedded Minimal Surfaces, Exotic Spheres, and Manifolds with Positive Ricci Curvature
Annals of Mathematics Embedded Minimal Surfaces, Exotic Spheres, and Manifolds with Positive Ricci Curvature Author(s): William Meeks III, Leon Simon and Shing-Tung Yau Source: Annals of Mathematics, Second Series, Vol. 116, No. 3 (Nov., 1982), pp. 621-659 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/2007026 Accessed: 06-02-2018 14:54 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics This content downloaded from 129.93.180.106 on Tue, 06 Feb 2018 14:54:09 UTC All use subject to http://about.jstor.org/terms Annals of Mathematics, 116 (1982), 621-659 Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature By WILLIAM MEEKS III, LEON SIMON and SHING-TUNG YAU Let N be a three dimensional Riemannian manifold. Let E be a closed embedded surface in N. Then it is a question of basic interest to see whether one can deform : in its isotopy class to some "canonical" embedded surface. From the point of view of geometry, a natural "canonical" surface will be the extremal surface of some functional defined on the space of embedded surfaces. -
Surfaces Minimales : Theorie´ Variationnelle Et Applications
SURFACES MINIMALES : THEORIE´ VARIATIONNELLE ET APPLICATIONS FERNANDO CODA´ MARQUES - IMPA (NOTES BY RAFAEL MONTEZUMA) Contents 1. Introduction 1 2. First variation formula 1 3. Examples 4 4. Maximum principle 5 5. Calibration: area-minimizing surfaces 6 6. Second Variation Formula 8 7. Monotonicity Formula 12 8. Bernstein Theorem 16 9. The Stability Condition 18 10. Simons' Equation 29 11. Schoen-Simon-Yau Theorem 33 12. Pointwise curvature estimates 38 13. Plateau problem 43 14. Harmonic maps 51 15. Positive Mass Theorem 52 16. Harmonic Maps - Part 2 59 17. Ricci Flow and Poincar´eConjecture 60 18. The proof of Willmore Conjecture 65 1. Introduction 2. First variation formula Let (M n; g) be a Riemannian manifold and Σk ⊂ M be a submanifold. We use the notation jΣj for the volume of Σ. Consider (x1; : : : ; xk) local coordinates on Σ and let @ @ gij(x) = g ; ; for 1 ≤ i; j ≤ k; @xi @xj 1 2 FERNANDO CODA´ MARQUES - IMPA (NOTES BY RAFAEL MONTEZUMA) be the components of gjΣ. The volume element of Σ is defined to be the p differential k-form dΣ = det gij(x). The volume of Σ is given by Z Vol(Σ) = dΣ: Σ Consider the variation of Σ given by a smooth map F :Σ × (−"; ") ! M. Use Ft(x) = F (x; t) and Σt = Ft(Σ). In this section we are interested in the first derivative of Vol(Σt). Definition 2.1 (Divergence). Let X be a arbitrary vector field on Σk ⊂ M. We define its divergence as k X (1) divΣX(p) = hrei X; eii; i=1 where fe1; : : : ; ekg ⊂ TpΣ is an orthonormal basis and r is the Levi-Civita connection with respect to the Riemannian metric g. -
Quintic Parametric Polynomial Minimal Surfaces and Their Properties
Differential Geometry and its Applications 28 (2010) 697–704 Contents lists available at ScienceDirect Differential Geometry and its Applications www.elsevier.com/locate/difgeo Quintic parametric polynomial minimal surfaces and their properties ∗ Gang Xu a,c, ,Guo-zhaoWangb a Hangzhou Dianzi University, Hangzhou 310018, PR China b Department of Mathematics, Zhejiang University, Hangzhou 310027, PR China c Galaad, INRIA Sophia-Antipolis, 2004 Route des Lucioles, 06902 Sophia-Antipolis Cedex, France article info abstract Article history: In this paper, quintic parametric polynomial minimal surface and their properties are Received 28 November 2009 discussed. We first propose the sufficient condition of quintic harmonic polynomial Received in revised form 14 April 2010 parametric surface being a minimal surface. Then several new models of minimal surfaces Available online 31 July 2010 with shape parameters are derived from this condition. We also study the properties Communicated by Z. Shen of new minimal surfaces, such as symmetry, self-intersection on symmetric planes and MSC: containing straight lines. Two one-parameter families of isometric minimal surfaces are 53A10 also constructed by specifying some proper shape parameters. 49Q05 © 2010 Elsevier B.V. All rights reserved. 49Q10 Keywords: Minimal surface Harmonic surfaces Isothermal parametric surface Isometric minimal surfaces 1. Introduction Since Lagrange derived the minimal surface equation in R3 in 1762, minimal surfaces have a long history of over 200 years. A minimal surface is a surface with vanishing mean curvature. As the mean curvature is the variation of area functional, minimal surfaces include the surfaces minimizing the area with a fixed boundary. Because of their attractive properties, minimal surfaces have been extensively employed in many areas such as architecture, material science, aviation, ship manufacture, biology and so on. -
Generating Carbon Schwarzites Via Zeolite-Templating
Generating carbon schwarzites via zeolite-templating Efrem Brauna, Yongjin Leeb,c, Seyed Mohamad Moosavib, Senja Barthelb, Rocio Mercadod, Igor A. Baburine, Davide M. Proserpiof,g, and Berend Smita,b,1 aDepartment of Chemical and Biomolecular Engineering, University of California, Berkeley, CA 94720; bInstitut des Sciences et Ingenierie´ Chimiques (ISIC), Valais, Ecole´ Polytechnique Fed´ erale´ de Lausanne (EPFL), CH-1951 Sion, Switzerland; cSchool of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China; dDepartment of Chemistry, University of California, Berkeley, CA 94720; eTheoretische Chemie, Technische Universitat¨ Dresden, 01062 Dresden, Germany; fDipartimento di Chimica, Universita` degli Studi di Milano, 20133 Milan, Italy; and gSamara Center for Theoretical Materials Science (SCTMS), Samara State Technical University, Samara 443100, Russia Edited by Monica Olvera de la Cruz, Northwestern University, Evanston, IL, and approved July 9, 2018 (received for review March 25, 2018) Zeolite-templated carbons (ZTCs) comprise a relatively recent ZTC can be seen as a schwarzite (24–28). Indeed, the experi- material class synthesized via the chemical vapor deposition of mental properties of ZTCs are exactly those which have been a carbon-containing precursor on a zeolite template, followed predicted for schwarzites, and hence ZTCs are considered as by the removal of the template. We have developed a theoret- promising materials for the same applications (21, 22). As we ical framework to generate a ZTC model from any given zeolite will show, the similarity between ZTCs and schwarzites is strik- structure, which we show can successfully predict the structure ing, and we explore this similarity to establish that the theory of known ZTCs. We use our method to generate a library of of schwarzites/TPMSs is a useful concept to understand ZTCs. -
Minimal Surfaces Saint Michael’S College
MINIMAL SURFACES SAINT MICHAEL’S COLLEGE Maria Leuci Eric Parziale Mike Thompson Overview What is a minimal surface? Surface Area & Curvature History Mathematics Examples & Applications http://www.bugman123.com/MinimalSurfaces/Chen-Gackstatter-large.jpg What is a Minimal Surface? A surface with mean curvature of zero at all points Bounded VS Infinite A plane is the most trivial minimal http://commons.wikimedia.org/wiki/File:Costa's_Minimal_Surface.png surface Minimal Surface Area Cube side length 2 Volume enclosed= 8 Surface Area = 24 Sphere Volume enclosed = 8 r=1.24 Surface Area = 19.32 http://1.bp.blogspot.com/- fHajrxa3gE4/TdFRVtNB5XI/AAAAAAAAAAo/AAdIrxWhG7Y/s160 0/sphere+copy.jpg Curvature ~ rate of change More Curvature dT κ = ds Less Curvature History Joseph-Louis Lagrange first brought forward the idea in 1768 Monge (1776) discovered mean curvature must equal zero Leonhard Euler in 1774 and Jean Baptiste Meusiner in 1776 used Lagrange’s equation to find the first non- trivial minimal surface, the catenoid Jean Baptiste Meusiner in 1776 discovered the helicoid Later surfaces were discovered by mathematicians in the mid nineteenth century Soap Bubbles and Minimal Surfaces Principle of Least Energy A minimal surface is formed between the two boundaries. The sphere is not a minimal surface. http://arxiv.org/pdf/0711.3256.pdf Principal Curvatures Measures the amount that surfaces bend at a certain point Principal curvatures give the direction of the plane with the maximal and minimal curvatures http://upload.wikimedia.org/wi -
Schwarz' P and D Surfaces Are Stable
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Differential Geometry and its Applications 2 (1992) 179-195 179 North-Holland Schwarz’ P and D surfaces are stable Marty Ross Department of Mathematics, Rice University, Houston, TX 77251, U.S.A. Communicated by F. J. Almgren, Jr. Received 30 July 1991 Ross, M., Schwarz’ P and D surfaces are stable, Diff. Geom. Appl. 2 (1992) 179-195. Abstract: The triply-periodic minimal P surface of Schwarz is investigated. It is shown that this surface is stable under periodic volume-preserving variations. As a consequence, the conjugate D surface is also stable. Other notions of stability are discussed. Keywords: Minimal surface, stable, triply-periodic, superfluous. MS classijication: 53A. 1. Introduction The existence of naturally occuring periodic phenomena is of course well known, Recently there have been suggestions that certain periodic behavior, including surface interfaces, might be suitably modelled by triply-periodic minimal and constant mean curvature surfaces (see [18, p. 2401, [l], [2] and th e references cited there). This is not unreasonable as such surfaces are critical points for a simple functional, the area functional. However, a naturally occuring surface ipso facto exhibits some form of stability, and thus we are led to consider as well the second variation of any candidate surface. Given an oriented immersed surface M = z(S) 5 Iw3, we can consider a smooth variation Mt = z(S,t) of M, and the resulting effect upon the area, A(t). M = MO is minimal if A’(0) = 0 for every compactly supported variation which leaves dM fixed. -
MINIMAL SURFACES THAT GENERALIZE the ENNEPER's SURFACE 1. Introduction
Novi Sad J. Math. Vol. 40, No. 2, 2010, 17-22 MINIMAL SURFACES THAT GENERALIZE THE ENNEPER'S SURFACE Dan Dumitru1 Abstract. In this article we give the construction of some minimal surfaces in the Euclidean spaces starting from the Enneper's surface. AMS Mathematics Subject Classi¯cation (2000): 53A05 Key words and phrases: Enneper's surface, minimal surface, curvature 1. Introduction We start with the well-known Enneper's surface which is de¯ned in the 2 3 u3 3-dimensional Cartesian by the application x : R ! R ; x(u; v) = (u ¡ 3 + 2 v3 2 2 2 uv ; v¡ 3 +vu ; u ¡v ): The point (u; v) is mapped to (f(u; v); g(u; v); h(u; v)), which is a point on Enneper's surface [2]. This surface has self-intersections and it is one of the ¯rst examples of minimal surface. The fact that Enneper's surface is minimal is a consequence of the following theorem: u3 2 Theorem 1.1. [1] Consider the Enneper's surface x(u; v) = (u ¡ 3 + uv ; v ¡ v3 2 2 2 3 + vu ; u ¡ v ): Then: a) The coe±cients of the ¯rst fundamental form are (1.1) E = G = (1 + u2 + v2)2;F = 0: b) The coe±cients of the second fundamental form are (1.2) e = 2; g = ¡2; f = 0: c) The principal curvatures are 2 2 (1.3) k = ; k = ¡ : 1 (1 + u2 + v2)2 2 (1 + u2 + v2)2 Eg+eG¡2F f It follows that the mean curvature H = EG¡F 2 is zero, which means the Enneper's surface is minimal. -
Minimal Surfaces, a Study
Alma Mater Studiorum · Università di Bologna SCUOLA DI SCIENZE Corso di Laurea in Matematica MINIMAL SURFACES, A STUDY Tesi di Laurea in Geometria Differenziale Relatore: Presentata da: Dott.ssa CLAUDIA MAGNAPERA ALESSIA CATTABRIGA VI Sessione Anno Accademico 2016/17 Abstract Minimal surfaces are of great interest in various fields of mathematics, and yet have lots of application in architecture and biology, for instance. It is possible to list different equivalent definitions for such surfaces, which correspond to different approaches. In the following thesis we will go through some of them, concerning: having mean cur- vature zero, being the solution of Lagrange’s partial differential equation, having the harmonicity property, being the critical points of the area functional, being locally the least-area surface with a given boundary and proving the existence of a solution of the Plateau’s problem. Le superfici minime, sono di grande interesse in vari campi della matematica, e parec- chie sono le applicazioni in architettura e in biologia, ad esempio. È possibile elencare diverse definizioni equivalenti per tali superfici, che corrispondono ad altrettanti approcci. Nella seguente tesi ne affronteremo alcuni, riguardanti: la curvatura media, l’equazione differenziale parziale di Lagrange, la proprietà di una funzione di essere armonica, i punti critici del funzionale di area, le superfici di area minima con bordo fissato e la soluzione del problema di Plateau. 2 Introduction The aim of this thesis is to go through the classical minimal surface theory, in this specific case to state six equivalent definitions of minimal surface and to prove the equiv- alence between them. -
Deformations of the Gyroid and Lidinoid Minimal Surfaces
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES ADAM G. WEYHAUPT Abstract. The gyroid and Lidinoid are triply periodic minimal surfaces of genus 3 embed- ded in R3 that contain no straight lines or planar symmetry curves. They are the unique embedded members of the associate families of the Schwarz P and H surfaces. In this paper, we prove the existence of two 1-parameter families of embedded triply periodic minimal surfaces of genus 3 that contain the gyroid and a single 1-parameter family that contains the Lidinoid. We accomplish this by using the flat structures induced by the holomorphic 1 1-forms Gdh, G dh, and dh. An explicit parametrization of the gyroid using theta functions enables us to find a curve of solutions in a two-dimensional moduli space of flat structures by means of an intermediate value argument. Contents 1. Introduction 2 2. Preliminaries 3 2.1. Parametrizing minimal surfaces 3 2.2. Cone metrics 5 2.3. Conformal quotients of triply periodic minimal surfaces 5 3. Parametrization of the gyroid and description of the periods 7 3.1. The P Surface and tP deformation 7 3.2. The period problem for the P surface 11 3.3. The gyroid 14 4. Proof of main theorem 16 4.1. Sketch of the proof 16 4.2. Horizontal and vertical moduli spaces for the tG family 17 4.3. Proof of the tG family 20 5. The rG and rL families 25 5.1. Description of the Lidinoid 25 5.2. Moduli spaces for the rL family 29 5.3. -
Complete Minimal Surfaces in R3
Publicacions Matem`atiques, Vol 43 (1999), 341–449. COMPLETE MINIMAL SURFACES IN R3 Francisco J. Lopez´ ∗ and Francisco Mart´ın∗ Abstract In this paper we review some topics on the theory of complete minimal surfaces in three dimensional Euclidean space. Contents 1. Introduction 344 1.1. Preliminaries .......................................... 346 1.1.1. Weierstrass representation .......................346 1.1.2. Minimal surfaces and symmetries ................349 1.1.3. Maximum principle for minimal surfaces .........349 1.1.4. Nonorientable minimal surfaces ..................350 1.1.5. Classical examples ...............................351 2. Construction of minimal surfaces with polygonal boundary 353 3. Gauss map of minimal surfaces 358 4. Complete minimal surfaces with bounded coordinate functions 364 5. Complete minimal surfaces with finite total curvature 367 5.1. Existence of minimal surfaces of least total curvature ...370 5.1.1. Chen and Gackstatter’s surface of genus one .....371 5.1.2. Chen and Gackstatter’s surface of genus two .....373 5.1.3. The surfaces of Espirito-Santo, Thayer and Sato..378 ∗Research partially supported by DGICYT grant number PB97-0785. 342 F. J. Lopez,´ F. Mart´ın 5.2. New families of examples...............................382 5.3. Nonorientable examples ................................388 5.3.1. Nonorientable minimal surfaces of least total curvature .......................................391 5.3.2. Highly symmetric nonorientable examples ........396 5.4. Uniqueness results for minimal surfaces of least total curvature ..............................................398 6. Properly embedded minimal surfaces 407 6.1. Examples with finite topology and more than one end ..408 6.1.1. Properly embedded minimal surfaces with three ends: Costa-Hoffman-Meeks and Hoffman-Meeks families..........................................409 6.1.2. -
Experimental Form-Finding for Möbius Strip and Ennerper Minimal
ISSN: 2319-5967 ISO 9001:2008 Certified International Journal of Engineering Science and Innovative Technology (IJESIT) Volume 2, Issue 5, September 2013 Experimental Form-Finding for Möbius Strip and Ennerper Minimal Surfaces Using Soap Film Models Hooi Min YEE1, Kok Keong CHOONHG2, Jae-Yeol KIM3 1Faculty of Civil Engineering, University Technology MARA Penang, Malaysia 2Academic staff , School of Civil Engineering, University Sains Malaysia, Penang, Malaysia 3Corresponding Author, Associate Professor, Hyupsung University, Hwasung, Korea Abstract—This paper describes an effort to produce soap film models with boundaries corresponding to those defined by Möbius strip and Enneper minimal surfaces. Soap film models with surfaces deviating from mathematically defined minimal surfaces of Möbius strip and Enneper are shown. Index Terms— Form-finding, Soap film model, Möbius strip, Enneper. I. INTRODUCTION Soap film model is used to find the surface form that minimizes area subject to appreciate constraints. Ireland [1] has presented some puzzling measurements of the tension between soap film and a solid surface. Moulton and Pelesko[2] have investigated Catenoid soap film model subjected to an axially symmetric electric field. The experimental and theoretical analysis for this Field Driven Mean Curvature Surface (FDMC) surface provides a step in understanding how electric fields interact with surfaces driven by surface tension. Such interactions may be of great benefit in micro scale systems such as micro electro mechanical systems (MEMS) and self-assembly. Brakke[3] has treated the area minimization problem with boundary constraints. Koiso and Palmer[4] have studied a variational problem whose solutions are a geometric model for thin films with gravity which is partially supported by a given contour.