Properly Embedded Minimal Surfaces with Finite Topology
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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. -
A Chiral Family of Triply-Periodic Minimal Surfaces from the Quartz Network
A chiral family of triply-periodic minimal surfaces from the quartz network Shashank G. Markande 1, Matthias Saba2, G. E. Schroder-Turk¨ 3 and Elisabetta A. Matsumoto 1∗ 1School of Physics, Georgia Tech, Atlanta, GA 30318, USA 2 Department of Physics, Imperial College, London, UK 3School of Engineering and IT, Mathematics and Statistics, Murdoch University, Murdoch, Australia ∗ Author for correspondence: [email protected] May 21, 2018 Abstract We describe a new family of triply-periodic minimal surfaces with hexagonal symmetry, related to the quartz (qtz) and its dual (the qzd net). We provide a solution to the period problem and provide a parametrisation of these surfaces, that are not in the regular class, by the Weierstrass-Enneper formalism. We identified this analytical description of the surface by generating an area-minimising mesh interface from a pair of dual graphs (qtz & qzd) using the generalised Voronoi construction of De Campo, Hyde and colleagues, followed by numerical identification of the flat point structure. This mechanism is not restricted to the specific pair of dual graphs, and should be applicable to a broader set of possible dual graph topologies and their corresponding minimal surfaces, if existent. 1 Introduction The discovery of cubic structures with vivid photonic properties in a wide range of biological systems, such as those responsible for the structural colouring seen in the wing scales of butterflies [1, 2, 3, 4] and in beetle cuticles [5, 6], has driven a renewed interest in mathematics of minimal surfaces – surfaces which, every- where, have zero mean curvature. The optical properties in these structures comes from the periodic nature of their bulk structure. -
Stable Constant Mean Curvature Tori and the Isoperimetric Problem in Three Space Forms, Comment
j. differential geometry 74 (2006) 69-92 ONE-SIDED COMPLETE STABLE MINIMAL SURFACES Antonio Ros Abstract We prove that there are no complete one-sided stable minimal surfaces in the Euclidean 3-space. We classify least area surfaces in the quotient of R3 by one or two linearly independent translations and we give sharp upper bounds of the genus of compact two- sided index one minimal surfaces in non-negatively curved ambient spaces. Finally we estimate from below the index of complete minimal surfaces in flat spaces in terms of the topology of the surface. 1. Introduction Stable minimal surfaces in 3-manifolds minimize area up to second order for any compactly supported smooth deformation. They play a fundamental role in minimal surface theory. Do Carmo and Peng [6], Fischer-Colbrie and Schoen [11] and Pogorelov [26], independently, proved that complete stable two-sided minimal surfaces immersed in the euclidean 3-space are flat. The results of [11] apply to 3-manifolds with nonnegative scalar curvature. In these ambient spaces, Fischer-Colbrie [10] proved that two-sided complete stable minimal surfaces with com- pact boundary have finite finite total curvature. Recently Meeks and Rosenberg [19] classified two-sided stable complete minimal surfaces in 3-manifolds of the type M R, M being a closed Riemannian surface. In the one-sided case the× stability condition (and the area minimiza- tion problem in general) becomes more subtle. This case should not be considered as a minor or secondary one. On the contrary, it presents a rich and interesting geometry, even in flat or positively curved 3- manifolds, which is far from being understood. -
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 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. -
Classical Minimal Surfaces in Euclidean Space by Examples
Classical Minimal Surfaces in Euclidean Space by Examples Geometric and Computational Aspects of the Weierstra¼ representation September 25, 2001 Matthias Weber 2 Contents 1 Introduction 6 2 Basic Di®erential Geometry Of Surfaces 7 2.1 Local Surface Parameterizations . 7 2.2 The Stereographic Projection . 8 2.3 The Normal Vector or Gau¼ Map . 9 3 The Weierstrass Representation 11 3.1 The Gau¼ map . 11 3.2 Harmonic coordinate functions . 12 3.3 Weierstrass data . 13 3.4 The ¯rst examples . 15 3.5 Local analysis at regular points . 17 3.6 The associate family . 19 3.7 The conjugate surface and symmetries . 20 4 Minimal surfaces on Punctured Spheres 22 4.1 Period conditions . 22 4.2 A surface with a planar end and an Enneper end . 23 4.3 A sphere with one planar and two catenoid ends . 24 4.4 The k-Noids of Jorge and Meeks . 25 5 The Scherk Surfaces 28 5.1 The singly periodic Scherk surface . 28 3 5.2 The period conditions . 28 5.3 Changing the angle between the ends . 30 5.4 The doubly periodic Scherk surface . 32 5.5 Singly periodic Scherk surfaces with higher dihedral symmetry 34 5.6 The twist deformation of Scherk's singly periodic surface . 35 5.7 The Schwarz-Christo®el formula . 39 6 Minimal Surfaces De¯ned On Punctured Tori 41 6.1 Complex tori . 41 6.2 Algebraic equations . 42 6.3 Theta functions . 43 6.4 Elliptic functions via Schwarz-Christo®el maps . 44 6.5 The Chen-Gackstatter surface . -
What Are Minimal Surfaces?
What are Minimal surfaces? RUKMINI DEY International Center for Theoretical Sciences (ICTS-TIFR), Bangalore ICTS-TIFR, Bangalore May, 2018 Topology of compact oriented surfaces – genus Sphere, torus, double torus are usual names you may have heard. Sphere and ellipsoids are abundant in nature – for example a ball and an egg. They are topologically “same” – in the sense if they were made out of plasticine, you could have deformed one to the other. Figure 1: Sphere and a glass ball Rukmini Dey µ¶ ToC ·¸ Minimal Surfaces page 2 of 46 Figure 2: Ellipsoids and an egg Figure 3: Tori and donut Rukmini Dey µ¶ ToC ·¸ Minimal Surfaces page 3 of 46 Figure 4: Triple torus and surface of a pretzel Rukmini Dey µ¶ ToC ·¸ Minimal Surfaces page 4 of 46 Handle addition to the sphere –genus Figure 5: Triple torus In fact as the above figure suggests one can construct more and more complicated sur- faces by adding handles to a sphere – the number of handles added is the genus. Rukmini Dey µ¶ ToC ·¸ Minimal Surfaces page 5 of 46 Genus classifies compact oriented surfaces topologically Sphere (genus g =0) is not topologically equivalent to torus (genus g =1) and none of them are topologically equivalent to double torus (genus g =2) and so on. One cannot continuously deform one into the other if the genus-es are different. F Euler’s formula: Divide the surface up into faces (without holes) with a certain number of vertices and edges. Let no. of vertices = V , no. of faces = F , no. of edges = E, no. -
Stratified Multivarifolds, and the Plateau Problem
Minimal Surfaces, Stratified Multivarifolds, and the Plateau Problem BA0 TRONG THI A. T. FOMENKO TRANSLATIONS OF MATHEMATICAL MONOGRAPHS TRANSLATIONS OF MATHEMATICAL MONOGRAPHS VOLUME84 Minimal Surfaces, Stratified M u ltiva rifo lds, and the Plateau Problem 8AO TRONG THI A. T. FOMENKO American Mathematical Society ProvidenceRhode island AAO 'IOHI' TXH A. T. 4)OMEHKO MHHHMAJIbHbIE IIOBEPXHOCTH H HPOEJIEMA HJIATO «HAYKA», MOCKBA, 1987 Translated from the Russian by E. J. F. Primrose Translation edited by Ben Silver 1980 Mathematics Subject Classification (1985 Revision). Primary49F10, 53A10; Secondary 58E12, 58E20. ABSTRACT. The book is an account of the current state of the theory of minimal surfaces and of one of the most important chapters of this theory-the Plateau problem, i.e. the problem of the existence of a minimal surface with boundary prescribed in advance. The authors exhibit deep connections of minimal surface theory with differential equations, Lie groups and Lie algebras, topology, and multidimensional variational calculus. The presentation is simplified to a large extent; the book is furnished with a wealth of illustrative material. Bibliography: 471 titles. 153 figures, 2 tables. Library of Congress Cataloging-ln-Publicatlon Data Dao, Trong Thi. [Minimal'nye poverkhnosti i problema Plato. English] Minimal surfaces, stratified multivarifolds, and the Plateau problem/Dio Trong Thi, A. T. Fomenko. p.cm. - (Translations of mathematical monographs; v. 84) Translation of: Minimal'nye poverkhnosti i problema Plato. Includes bibliographical references and index. ISBN 0-8218-4536-5 1. Surfaces, Minimal.2. Plateau's problem.I. Fomenko, A. T.II. Title.III. Series. QA644.D2813 1991 90-22932 516.3'62-dc2O CIP Copyright ©1991 by the American Mathematical Society. -
Minimal Surfaces for Architectural Constructions Udc 72.01(083.74)(045)=111
FACTA UNIVERSITATIS Series: Architecture and Civil Engineering Vol. 6, No 1, 2008, pp. 89 - 96 DOI: 10.2298/FUACE0801089V MINIMAL SURFACES FOR ARCHITECTURAL CONSTRUCTIONS UDC 72.01(083.74)(045)=111 Ljubica S. Velimirović1, Grozdana Radivojević2, Mića S. Stanković2, Dragan Kostić1 1 University of Niš, Faculty of Science and Mathematics, Niš, Serbia 2University of Niš, The Faculty of Civil Engineering and Architecture, Serbia Abstract. Minimal surfaces are the surfaces of the smallest area spanned by a given boundary. The equivalent is the definition that it is the surface of vanishing mean curvature. Minimal surface theory is rapidly developed at recent time. Many new examples are constructed and old altered. Minimal area property makes this surface suitable for application in architecture. The main reasons for application are: weight and amount of material are reduced on minimum. Famous architects like Otto Frei created this new trend in architecture. In recent years it becomes possible to enlarge the family of minimal surfaces by constructing new surfaces. Key words: Minimal surface, spatial roof surface, area, soap film, architecture 1. INTRODUCTION If we dip a metal wire-closed space curve into a soap solution, when we pull it out, a soap film forms. A nature solves a mathematical question of finding a surface of the least surface area for a given boundary. Among all possible surfaces soap film finds one with the least surface area. Deep mathematical problems lie in the theory behind. The theory of minimal surfaces is a branch of mathematics that has been intensively developed, particularly recently. On the base of this theory we can investigate membranes in living cells, capillary phenomena, polymer chemistry, crystallography. -
New Families of Embedded Triply Periodic Minimal Surfaces of Genus Three in Euclidean Space
Fontbonne University GriffinShare All Theses, Dissertations, and Capstone Projects Theses, Dissertations, and Capstone Projects 2006 New Families of Embedded Triply Periodic Minimal Surfaces of Genus Three in Euclidean Space Adam G. Weyhaupt Indiana University Follow this and additional works at: https://griffinshare.fontbonne.edu/all-etds Part of the Mathematics Commons This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License. Recommended Citation Weyhaupt, Adam G., "New Families of Embedded Triply Periodic Minimal Surfaces of Genus Three in Euclidean Space" (2006). All Theses, Dissertations, and Capstone Projects. 160. https://griffinshare.fontbonne.edu/all-etds/160 NEW FAMILIES OF EMBEDDED TRIPLY PERIODIC MINIMAL SURFACES OF GENUS THREE IN EUCLIDEAN SPACE Adam G. Weyhaupt Submitted to the faculty of the University Graduate School in partial fulfillment of the requirements for the degree Doctor of Philosophy in the Department of Mathematics Indiana University August 2006 Accepted by the Graduate Faculty, Indiana University, in partial fulfillment of the require- ments for the degree of Doctor of Philosophy. Matthias Weber, Ph.D. Jiri Dadok, Ph.D. Bruce Solomon, Ph.D. Peter Sternberg, Ph.D. 4 August 2006 ii Copyright 2006 Adam G. Weyhaupt ALL RIGHTS RESERVED iii To Julia — my source of strength, my constant companion, my love forever and to Brandon and Ryan — whose exuberance and smiles lift my spirits daily and have changed my life in so many wonderful ways. iv Acknowledgements I can not imagine having completed graduate school with a different advisor: thank you, Matthias Weber, for your encouragement, your guidance, your constant availability, and for making me feel like a colleague. -
What Are Minimal Surfaces?
What are minimal surfaces? RUKMINI DEY International Center for Theoretical Sciences (ICTS-TIFR), Bangalore SWSM, ICTS, Bangalore May , 2019 Topology of compact oriented surfaces – genus Sphere, torus, double torus are usual names you may have heard. Sphere and ellipsoids are abundant in nature – for example a ball and an egg. They are topologically “same” – in the sense if they were made out of plasticine, you could have deformed one to the other. Figure 1: Sphere and a glass ball Rukmini Dey µ¶ ToC ·¸ Minimal Surfaces page 2 of 54 Figure 2: Ellipsoids and an egg Figure 3: Tori and donut Rukmini Dey µ¶ ToC ·¸ Minimal Surfaces page 3 of 54 Figure 4: Triple torus and surface of a pretzel Rukmini Dey µ¶ ToC ·¸ Minimal Surfaces page 4 of 54 Handle addition to the sphere –genus Figure 5: Triple torus In fact as the above figure suggests one can construct more and more complicated sur- faces by adding handles to a sphere – the number of handles added is the genus. Rukmini Dey µ¶ ToC ·¸ Minimal Surfaces page 5 of 54 Genus classifies compact oriented surfaces topologically Sphere (genus g =0) is not topologically equivalent to torus (genus g =1) and none of them are topologically equivalent to double torus (genus g =2) and so on. One cannot continuously deform one into the other if the genus-es are different. F Euler’s formula: Divide the surface up into faces (without holes) with a certain number of vertices and edges. Let no. of vertices = V , no. of faces = F , no. of edges = E, no. -
Collected Atos
Mathematical Documentation of the objects realized in the visualization program 3D-XplorMath Select the Table Of Contents (TOC) of the desired kind of objects: Table Of Contents of Planar Curves Table Of Contents of Space Curves Surface Organisation Go To Platonics Table of Contents of Conformal Maps Table Of Contents of Fractals ODEs Table Of Contents of Lattice Models Table Of Contents of Soliton Traveling Waves Shepard Tones Homepage of 3D-XPlorMath (3DXM): http://3d-xplormath.org/ Tutorial movies for using 3DXM: http://3d-xplormath.org/Movies/index.html Version November 29, 2020 The Surfaces Are Organized According To their Construction Surfaces may appear under several headings: The Catenoid is an explicitly parametrized, minimal sur- face of revolution. Go To Page 1 Curvature Properties of Surfaces Surfaces of Revolution The Unduloid, a Surface of Constant Mean Curvature Sphere, with Stereographic and Archimedes' Projections TOC of Explicitly Parametrized and Implicit Surfaces Menu of Nonorientable Surfaces in previous collection Menu of Implicit Surfaces in previous collection TOC of Spherical Surfaces (K = 1) TOC of Pseudospherical Surfaces (K = −1) TOC of Minimal Surfaces (H = 0) Ward Solitons Anand-Ward Solitons Voxel Clouds of Electron Densities of Hydrogen Go To Page 1 Planar Curves Go To Page 1 (Click the Names) Circle Ellipse Parabola Hyperbola Conic Sections Kepler Orbits, explaining 1=r-Potential Nephroid of Freeth Sine Curve Pendulum ODE Function Lissajous Plane Curve Catenary Convex Curves from Support Function Tractrix