Bour's Theorem in 4-Dimensional Euclidean
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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. -
Lord Kelvin's Error? an Investigation Into the Isotropic Helicoid
WesleyanUniversity PhysicsDepartment Lord Kelvin’s Error? An Investigation into the Isotropic Helicoid by Darci Collins Class of 2018 An honors thesis submitted to the faculty of Wesleyan University in partial fulfillment of the requirements for the Degree of Bachelor of Arts with Departmental Honors in Physics Middletown,Connecticut April,2018 Abstract In a publication in 1871, Lord Kelvin, a notable 19th century scientist, hypothesized the existence of an isotropic helicoid. He predicted that such a particle would be isotropic in drag and rotation translation coupling, and also have a handedness that causes it to rotate. Since this work was published, theorists have made predictions about the motion of isotropic helicoids in complex flows. Until now, no one has built such a particle or quantified its rotation translation coupling to confirm whether the particle has the properties that Lord Kelvin predicted. In this thesis, we show experimental, theoretical, and computational evidence that all conclude that Lord Kelvin’s geometry of an isotropic helicoid does not couple rotation and translation. Even in both the high and low Reynolds number regimes, Lord Kelvin’s model did not rotate through fluid. While it is possible there may be a chiral particle that is isotropic in drag and rotation translation coupling, this thesis presents compelling evidence that the geometry Lord Kelvin proposed is not one. Our evidence leads us to hypothesize that an isotropic helicoid does not exist. Contents 1 Introduction 2 1.1 What is an Isotropic Helicoid? . 2 1.1.1 Lord Kelvin’s Motivation . 3 1.1.2 Mentions of an Isotropic Helicoid in Literature . -
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 -
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. -
[Math.DG] 12 Nov 1996 Le Osrcsa Constructs Elser H Ufc Noisl
Mixing Materials and Mathematics∗ David Hoffman † Mathematical Sciences Research Institute, Berkeley CA 94720 USA August 3, 2018 “Oil and water don’t mix” says the old saw. But a variety of immiscible liquids, in the presence of a soap or some other surfactant, can self-assemble into a rich variety of regular mesophases. Characterized by their “inter- material dividing surfaces”—where the different substances touch— these structures also occur in microphase-separated block copolymers. The un- derstanding of the interface is key to prediction of material properties, but at present the relationship between the curvature of the dividing surfaces and the relevant molecular and macromolecular physics is not well understood. Moreover, there is only a partial theoretical understanding of the range of possible periodic surfaces that might occur as interfaces. Here, differential geometry, the mathematics of curved surfaces and their generalizations, is playing an important role in the experimental physics of materials. “Curved surfaces and chemical structures,” a recent issue of the Philosophical Trans- actions of the Royal Society of London 1 provides a good sample of current work. In one article called “A cubic Archimedean screw,” 2 the physicist Veit Elser constructs a triply periodic surface with cubic symmetry. (This means that a unit translation in any one of the three coordinate directions moves the surface onto itself.) See Figure 1. The singularities of the surface are arXiv:math/9611213v1 [math.DG] 12 Nov 1996 ∗A version of this article will appear in NATURE, November 7, 1996, under the title “A new turn for Archimedes.” †Supported by research grant DE-FG03-95ER25250 of the Applied Mathematical Sci- ence subprogram of the Office of Energy Research, U.S. -
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. -
Minimal Surfaces in ${\Mathbb {R}}^{4} $ Foliated by Conic Sections
MINIMAL SURFACES IN R4 FOLIATED BY CONIC SECTIONS AND PARABOLIC ROTATIONS OF HOLOMORPHIC NULL CURVES IN C4 HOJOO LEE 4 ABSTRACT. Using the complex parabolic rotations of holomorphic null curves in C , we transform minimal surfaces in Euclidean space R3 ⊂ R4 to a family of degenerate minimal surfaces in Eu- clidean space R4. Applying our deformation to holomorphic null curves in C3 ⊂ C4 induced by helicoids in R3, we discover new minimal surfaces in R4 foliated by conic sections with eccentricity grater than 1: hyperbolas or straight lines. Applying our deformation to holomorphic null curves in C3 induced by catenoids in R3, we can rediscover the Hoffman-Osserman catenoids in R4 foli- ated by conic sections with eccentricity smaller than 1: ellipses or circles. We prove the existence of minimal surfaces in R4 foliated by ellipses, which converge to circles at infinity. We construct minimal surfaces in R4 foliated by parabolas: conic sections which have eccentricity 1. To the memory of Professor Ahmad El Soufi 1. INTRODUCTION It is a fascinating fact that any simply connected minimal surface in R3 admits a 1-parameter family of minimal isometric deformations, called the associate family. Rotating the holomorphic null curve in C3 induced by the given minimal surface in R3 realizes such isometric deformation of minimal surfaces. For instance, catenoids (foliated by circles) belong to the associate families of helicoids (foliated by lines) in R3. R. Schoen [29] characterized catenoids as the only complete, embedded minimal surfaces in R3 with finite topology and two ends. F. L´opez and A. Ros [13] used the so called L´opez- Ros deformation (See Example 2.4) of holomorphic null curves in C3 to prove that planes and catenoids are the only embedded complete minimal surfaces in R3 with finite total curvature and genus zero. -
Minimal Surfaces
Minimal Surfaces December 13, 2012 Alex Verzea 260324472 MATH 580: Partial Dierential Equations 1 Professor: Gantumur Tsogtgerel 1 Intuitively, a Minimal Surface is a surface that has minimal area, locally. First, we will give a mathematical denition of the minimal surface. Then, we shall give some examples of Minimal Surfaces to gain a mathematical under- standing of what they are and nally move on to a generalization of minimal surfaces, called Willmore Surfaces. The reason for this is that Willmore Surfaces are an active and important eld of study in Dierential Geometry. We will end with a presentation of the Willmore Conjecture, which has recently been proved and with some recent work done in this area. Until we get to Willmore Surfaces, we assume that we are in R3. Denition 1: The two Principal Curvatures, k1 & k2 at a point p 2 S, 3 S⊂ R are the eigenvalues of the shape operator at that point. In classical Dierential Geometry, k1 & k2 are the maximum and minimum of the Second Fundamental Form. The principal curvatures measure how the surface bends by dierent amounts in dierent directions at that point. Below is a saddle surface together with normal planes in the directions of principal curvatures. Denition 2: The Mean Curvature of a surface S is an extrinsic measure of curvature; it is the average of it's two principal curvatures: 1 . H ≡ 2 (k1 + k2) Denition 3: The Gaussian Curvature of a point on a surface S is an intrinsic measure of curvature; it is the product of the principal curvatures: of the given point. -
Blackfolds, Plane Waves and Minimal Surfaces
Blackfolds, Plane Waves and Minimal Surfaces Jay Armas1;2 and Matthias Blau2 1 Physique Th´eoriqueet Math´ematique Universit´eLibre de Bruxelles and International Solvay Institutes ULB-Campus Plaine CP231, B-1050 Brussels, Belgium 2 Albert Einstein Center for Fundamental Physics, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland [email protected], [email protected] Abstract Minimal surfaces in Euclidean space provide examples of possible non-compact horizon ge- ometries and topologies in asymptotically flat space-time. On the other hand, the existence of limiting surfaces in the space-time provides a simple mechanism for making these configura- tions compact. Limiting surfaces appear naturally in a given space-time by making minimal surfaces rotate but they are also inherent to plane wave or de Sitter space-times in which case minimal surfaces can be static and compact. We use the blackfold approach in order to scan for possible black hole horizon geometries and topologies in asymptotically flat, plane wave and de Sitter space-times. In the process we uncover several new configurations, such as black arXiv:1503.08834v2 [hep-th] 6 Aug 2015 helicoids and catenoids, some of which have an asymptotically flat counterpart. In particular, we find that the ultraspinning regime of singly-spinning Myers-Perry black holes, described in terms of the simplest minimal surface (the plane), can be obtained as a limit of a black helicoid, suggesting that these two families of black holes are connected. We also show that minimal surfaces embedded in spheres rather than Euclidean space can be used to construct static compact horizons in asymptotically de Sitter space-times. -
Deformations of Triply Periodic Minimal Surfaces a Family of Gyroids
Deformations of Triply Periodic Minimal Surfaces A Family of Gyroids Adam G. Weyhaupt Department of Mathematics Indiana University Eastern Illinois University Colloquium Outline Minimal surfaces: definitions, examples, goal, and motivation Definitions and examples Goal and motivation The mathematical setting The Weierstraß Representation and Period Problem Cone metrics and the flat structures Sketch of the gyroid family Description of the P Surface Description of the gyroid Outline of Proof Homework Problems to Work On For More Information and Pictures Outline Minimal surfaces: definitions, examples, goal, and motivation Definitions and examples Goal and motivation The mathematical setting The Weierstraß Representation and Period Problem Cone metrics and the flat structures Sketch of the gyroid family Description of the P Surface Description of the gyroid Outline of Proof Homework Problems to Work On For More Information and Pictures Definition of a Minimal Surface Definition A minimal surface is a 2-dimensional surface in R3 with mean curvature H ≡ 0. Where does the name minimal come from? Let F : U ⊂ C → R3 parameterize a minimal surface; let d : U → R be smooth with compact support. Define a deformation of M by Fε : p 7→ F(p) + εd(p)N(p). d Area(Fε(U)) = 0 ⇐⇒ H ≡ 0 dε ε=0 Thus, “minimal surfaces” may really only be critical points for the area functional (but the name has stuck). Definition of a Minimal Surface Definition A minimal surface is a 2-dimensional surface in R3 with mean curvature H ≡ 0. Where does the name minimal come from? Let F : U ⊂ C → R3 parameterize a minimal surface; let d : U → R be smooth with compact support. -
Deformations of the Gyroid and Lidinoid Minimal Surfaces
Pacific Journal of Mathematics DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES ADAM G. WEYHAUPT Volume 235 No. 1 March 2008 PACIFIC JOURNAL OF MATHEMATICS Vol. 235, No. 1, 2008 DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES ADAM G. WEYHAUPT The gyroid and Lidinoid are triply periodic minimal surfaces of genus three embedded 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. We prove the existence of two 1-parameter families of embedded triply periodic minimal surfaces of genus three that contain the gyroid and a single 1-parameter family that contains the Lidi- noid. We accomplish this by using the flat structures induced by the holo- morphic 1-forms G dh, (1/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 intermedi- ate value argument. 1. Introduction The gyroid was discovered by Alan Schoen [1970], a NASA crystallographer in- terested in strong but light materials. Among its most curious properties was that, unlike other known surfaces at the time, the gyroid contains no straight lines or pla- nar symmetry curves [Karcher 1989; Große-Brauckmann and Wohlgemuth 1996]. Soon after, Bill Meeks [1975] discovered a 5-parameter family of embedded genus three triply periodic minimal surfaces. Theorem 1.1 [Meeks 1975]. There is a real five-dimensional family V of periodic hyperelliptic Riemann surfaces of genus three.