DOCSLIB.ORG

Conformal and Asymptotic Properties of Embedded Genus-G Minimal Surfaces with One End by Jacob Bernstein

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Conformal and Asymptotic Properties of Embedded Genus-G Minimal Surfaces with One End by Jacob Bernstein Conformal and Asymptotic Properties of Embedded Genus-g Minimal Surfaces with One End by Jacob Bernstein Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2009 c Jacob Bernstein, MMIX. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. Author............................................. ................. Department of Mathematics May 18, 2009 Certified by......................................... ................. Tobias H. Colding Professor of Mathematics Thesis Supervisor Accepted by......................................... ................ David Jerison Chairman, Department Committee on Graduate Students 2 Conformal and Asymptotic Properties of Embedded Genus-g Minimal Surfaces with One End by Jacob Bernstein Submitted to the Department of Mathematics on May 18, 2009, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Using the tools developed by Colding and Minicozzi in their study of the structure of embedded minimal surfaces in R3 [12, 19–22], we investigate the conformal and asymptotic properties of complete, embedded minimal surfaces of finite genus and one end. We first present a more geometric proof of the uniqueness of the helicoid than the original, due to Meeks and Rosenberg [45]. That is, the only properly embedded and complete minimal disks in R3 are the plane and the helicoid. We then extend these techniques to show that any complete, embedded minimal surface with one end and finite topology is conformal to a once-punctured compact Riemann surface. This completes the classification of the conformal type of embedded finite topology minimal surfaces in R3. Moreover, we show that such s surface has Weierstrass data asymptotic to that of the helicoid, and as a consequence is asymptotic to a helicoid (in a Hausdorff sense). As such, we call such surfaces genus-g helicoids. In addition, we sharpen results of Colding and Minicozzi on the shapes of embedded minimal disks in R3, giving a more precise scale on which minimal disks with large curvature are “helicoidal”. Finally, we begin to study the finer properties of the structure of genus-g helicoids, in particular showing that the space of genus-one helicoids is compact (after a suitably normalization). Thesis Supervisor: Tobias H. Colding Title: Professor of Mathematics 3 4 Acknowledgments I would like to thank my advisor, Tobias Colding, for giving me the problem that was the genesis of this thesis and for giving me support and help in navigating the vagaries of mathematical research. I would also like to thank Bill Minicozzi for his constant enthusiasm and for his always extremely helpful suggestions. There are many other people at MIT I would like to acknowledge (in no particular order): Richard Melrose for teaching me analysis and for humoring some of my more ill-posed queries; Hans Christianson and Brett Kotschwar for handling most of the rest; Linda Okun and the rest of the math department staff for helping me understand some of MIT’s more arcane policies; Chris Kottke being a great office-mate and for our many procrastination driven conversations; Gigliola Staffilani for helping me in my job search; Michael Eichmair for energizing geometric analysis at MIT and for being a great friend; Tom Mrowka and William Lopes for discussing some elementary Morse theory with me, the results of which are in Chapter 6; thanks also to Vedran, YankI, Lu, James and Max for many great mathematical discussions. Away from MIT, my collaborator Christine Breiner must be mentioned, both as a great conference co-attendee and friend. Special thanks to my fiance´eJessie Howell for putting up with me, even at my most slovenly and unshaven. I couldn’t have done it with out you. I would also like to thank my parents. I am especially grateful to my father exposing me to deep mathematics at young age, and thank him for his copy of Kelley’s General Topology, my first encounter with abstract mathematics. Finally, I would like to dedicate this thesis to the memory of Juha Heinonen without whom I very likely would have left mathematics. His contribution to my mathematical education cannot be overstated. 5 6 Contents Abstract 3 Acknowledgments 5 1 Introduction 11 2 Background 15 2.1 Minimal Surface Theory in R3 ...................... 15 2.1.1 Basictheory............................ 15 2.1.2 Classical and modern constructions . 18 2.2 Notation.................................. 20 3 Colding-Minicozzi Theory 23 3.1 Structure of Embedded Minimal Disks . 24 3.1.1 Pointsoflargecurvature . 24 3.1.2 Extendingthesheets . 26 3.1.3 Findinglargecurvature. 26 3.1.4 One-sidedcurvatureestimate . 27 3.2 SomeApplications ............................ 29 3.2.1 Laminationtheory ........................ 30 3.2.2 TheCalabi-Yauconjecture. 30 3.2.3 Generalizations to non-trivial topology . ... 31 4 Uniqueness of the Helicoid 33 4.1 MeeksandRosenberg’sApproach . 33 4.2 OutlineoftheArgument . .. .. 34 4.3 GeometricDecomposition . 35 4.3.1 Initialsheets............................ 35 4.3.2 Blow-uppairs ........................... 37 4.3.3 Asymptotichelicoids . 37 4.3.4 DecompositionofΣ. .. .. 38 4.4 Conformal Structure of a Complete Embedded Minimal Disk ..... 40 4.4.1 Conformalstructure . 40 4.4.2 Conformal mapping properties of the Gauss map . 40 4.4.3 Uniqueness ............................ 42 7 4.5 Addendum................................. 42 4.5.1 Blow-upsheets .......................... 43 4.5.2 Geometrynearablow-uppair . 43 5 Structure Near a Blow-up Pair 45 5.1 LipschitzApproximation . 45 5.2 ScaleoftheApproximation . 48 6 Genus-g Helicoids 51 6.1 OutlineoftheProof ........................... 52 6.2 GeometricDecomposition . 53 6.2.1 Structuralresults . .. .. 53 6.2.2 Blow-upsheets .......................... 54 6.2.3 Blow-Uppairs........................... 55 6.2.4 DecomposingΣ .......................... 56 6.3 ConformalStructureoftheend . 58 6.3.1 Winding number of the Gauss map . 58 6.3.2 Conformalstructureoftheend . 60 6.3.3 The proofs of Theorem 6.0.6 and Corollary 6.0.8 . .. 60 6.4 Addendum................................. 61 6.4.1 TopologicalstructureofΣ . 61 6.4.2 ProofsofProposition6.2.1and6.2.2 . 62 6.4.3 One-sidedcurvatureinΣ. 63 6.4.4 Geometric Bounds near blow-up pairs . 64 7 The Space of Genus-g Helicoids 65 7.1 OutlineofArgument ........................... 67 7.2 WeakCompactness ............................ 68 7.2.1 Technicallemmas ......................... 68 7.2.2 ProofofTheorem7.1.1. 71 7.3 The Intrinsic and Extrinsic Normalization . .... 74 7.3.1 Two-sidedboundsonthegenus . 75 7.3.2 Intrinsicnormalization . 78 7.3.3 Extrinsicnormalization. 79 7.4 Applications................................ 80 7.4.1 Compactness of (1, 1)...................... 80 E 7.4.2 Geometric Structure of (1, 1, R) ................ 84 E 8 Conclusion 87 8 List of Figures 2-1 Thecatenoid(Courtesy of MatthiasWeber) . .. 19 2-2 The helicoid (Courtesy of Matthias Weber) . ... 19 2-3 A genus one helicoid (Courtesy of Matthias Weber) . .... 20 3-1 Theone-sidedcurvatureestimate . 28 3-2 The one-sided curvature estimate in a cone . ... 29 5-1 A cross section of one of Meeks and Weber’s examples, with the axis as a circle. We indicate a subset which is a disk. Here R is the outer scale of said disk and s theblow-upscale. 46 5-2 A cross section of one of Colding and Minicozzi’s examples. We indicate the two important scales: R = 1 the outer scale and s the blow-up scale. (Here (0,s)isablow-uppair.)...................... 46 5-3 The points pi and ui. Note that the density ratio of ui is much larger than the density ratio of pi. ....................... 50 6-1 A rough sketch of the decomposition of Σ given by Theorem 6.1.1.. 53 6-2 Level curve examples in Proposition 6.3.2. (a) Initial orientation chosen at height x3 = h. (b) A curve pinching off from Ω1. (c) Two curves pinching from one. (d) A curve pinching off from Ω2. ......... 59 7-1 The points of interest in the proof of Lemma 7.2.7. .... 72 7-2 Illustrating the consequence of the one sided curvature estimates . 73 9 10 Chapter 1 Introduction The study of minimal surfaces has a long history, dating to the eighteenth century and the beginnings of the calculus of variations. The theory sits at a fundamental intersection of geometry, analysis and topology and has provided important tools, techniques and insights in all three areas. Moreover, even in its most classical setting, minimal surface theory remains an active area of research. Recall, a minimal surface is a surface that is a stationary point of the area functional; in other words, infinitesimal deformations of the surface do not change its area. A particularly important class of these, and indeed a major motivation for the theory, are surfaces that actually minimize area in a global sense, as these can be taken as a model of the shape of a soap film. While minimal surfaces can be studied in a large number of different contexts, we will restrict our attention to the classical setting of minimal surfaces in R3. This is, of course, the context in which the theory was originally developed and remains an area of active research. We will be interested in classifying the complete embedded minimal surfaces in R3. Before discussing such a classification program further, we first record the three most important such surfaces. We do this both to illustrate that this is a non-trivial question and to have some simple examples on hand. The first, and least interesting, is the plane, the second is the catenoid and the final is the helicoid. The catenoid was discovered by Euler in 1744 and is the surface of revolution of the catenary (see Figure 2-1). The helicoid was discovered by Meusnier in 1776 and looks like a double-spiral staircase (see Figure 2-2). It is the surface swept out by a line moving through space at a constant rate while rotating at a constant rate in the plane perpendicular to the motion.
Load more

Recommended publications
  • An Introduction to Topology the Classification Theorem for Surfaces by E
    An Introduction to Topology An Introduction to Topology The Classification theorem for Surfaces By E. C. Zeeman Introduction. The classification theorem is a beautiful example of geometric topology. Although it was discovered in the last century*, yet it manages to convey the spirit of present day research. The proof that we give here is elementary, and its is hoped more intuitive than that found in most textbooks, but in none the less rigorous. It is designed for readers who have never done any topology before. It is the sort of mathematics that could be taught in schools both to foster geometric intuition, and to counteract the present day alarming tendency to drop geometry. It is profound, and yet preserves a sense of fun. In Appendix 1 we explain how a deeper result can be proved if one has available the more sophisticated tools of analytic topology and algebraic topology. Examples. Before starting the theorem let us look at a few examples of surfaces. In any branch of mathematics it is always a good thing to start with examples, because they are the source of our intuition. All the following pictures are of surfaces in 3-dimensions. In example 1 by the word “sphere” we mean just the surface of the sphere, and not the inside. In fact in all the examples we mean just the surface and not the solid inside. 1. Sphere. 2. Torus (or inner tube). 3. Knotted torus. 4. Sphere with knotted torus bored through it. * Zeeman wrote this article in the mid-twentieth century. 1 An Introduction to Topology 5.
    [Show full text]
  • Equivelar Octahedron of Genus 3 in 3-Space
    Equivelar octahedron of genus 3 in 3-space Ruslan Mizhaev ([email protected]) Apr. 2020 ABSTRACT . Building up a toroidal polyhedron of genus 3, consisting of 8 nine-sided faces, is given. From the point of view of topology, a polyhedron can be considered as an embedding of a cubic graph with 24 vertices and 36 edges in a surface of genus 3. This polyhedron is a contender for the maximal genus among octahedrons in 3-space. 1. Introduction This solution can be attributed to the problem of determining the maximal genus of polyhedra with the number of faces - . As is known, at least 7 faces are required for a polyhedron of genus = 1 . For cases ≥ 8 , there are currently few examples. If all faces of the toroidal polyhedron are – gons and all vertices are q-valence ( ≥ 3), such polyhedral are called either locally regular or simply equivelar [1]. The characteristics of polyhedra are abbreviated as , ; [1]. 2. Polyhedron {9, 3; 3} V1. The paper considers building up a polyhedron 9, 3; 3 in 3-space. The faces of a polyhedron are non- convex flat 9-gons without self-intersections. The polyhedron is symmetric when rotated through 1804 around the axis (Fig. 3). One of the features of this polyhedron is that any face has two pairs with which it borders two edges. The polyhedron also has a ratio of angles and faces - = − 1. To describe polyhedra with similar characteristics ( = − 1) we use the Euler formula − + = = 2 − 2, where is the Euler characteristic. Since = 3, the equality 3 = 2 holds true.
    [Show full text]
  • INTRODUCTION to ALGEBRAIC GEOMETRY, CLASS 25 Contents 1
    INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 25 RAVI VAKIL Contents 1. The genus of a nonsingular projective curve 1 2. The Riemann-Roch Theorem with Applications but No Proof 2 2.1. A criterion for closed immersions 3 3. Recap of course 6 PS10 back today; PS11 due today. PS12 due Monday December 13. 1. The genus of a nonsingular projective curve The definition I’m going to give you isn’t the one people would typically start with. I prefer to introduce this one here, because it is more easily computable. Definition. The tentative genus of a nonsingular projective curve C is given by 1 − deg ΩC =2g 2. Fact (from Riemann-Roch, later). g is always a nonnegative integer, i.e. deg K = −2, 0, 2,.... Complex picture: Riemann-surface with g “holes”. Examples. Hence P1 has genus 0, smooth plane cubics have genus 1, etc. Exercise: Hyperelliptic curves. Suppose f(x0,x1) is a polynomial of homo- geneous degree n where n is even. Let C0 be the affine plane curve given by 2 y = f(1,x1), with the generically 2-to-1 cover C0 → U0.LetC1be the affine 2 plane curve given by z = f(x0, 1), with the generically 2-to-1 cover C1 → U1. Check that C0 and C1 are nonsingular. Show that you can glue together C0 and C1 (and the double covers) so as to give a double cover C → P1. (For computational convenience, you may assume that neither [0; 1] nor [1; 0] are zeros of f.) What goes wrong if n is odd? Show that the tentative genus of C is n/2 − 1.(Thisisa special case of the Riemann-Hurwitz formula.) This provides examples of curves of any genus.
    [Show full text]
  • Homeomorphisms of the 3-Sphere That Preserve a Heegaard Splitting of Genus Two
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 3, March 2008, Pages 1113–1123 S 0002-9939(07)09188-5 Article electronically published on November 30, 2007 HOMEOMORPHISMS OF THE 3-SPHERE THAT PRESERVE A HEEGAARD SPLITTING OF GENUS TWO SANGBUM CHO (Communicated by Daniel Ruberman) Abstract. Let H2 be the group of isotopy classes of orientation-preserving homeomorphisms of S3 that preserve a Heegaard splitting of genus two. In this paper, we construct a tree in the barycentric subdivision of the disk complex of a handlebody of the splitting to obtain a finite presentation of H2. 1. Introduction Let Hg be the group of isotopy classes of orientation-preserving homeomorphisms of S3 that preserve a Heegaard splitting of genus g,forg ≥ 2. It was shown by Goeritz [3] in 1933 that H2 is finitely generated. Scharlemann [7] gave a modern proof of Goeritz’s result, and Akbas [1] refined this argument to give a finite pre- sentation of H2. In arbitrary genus, first Powell [6] and then Hirose [4] claimed to have found a finite generating set for the group Hg, though serious gaps in both ar- guments were found by Scharlemann. Establishing the existence of such generating sets appears to be an open problem. In this paper, we recover Akbas’s presentation of H2 by a new argument. First, we define the complex P (V ) of primitive disks, which is a subcomplex of the disk complex of a handlebody V in a Heegaard splitting of genus two. Then we find a suitable tree T ,onwhichH2 acts, in the barycentric subdivision of P (V )toget a finite presentation of H2.
    [Show full text]
  • Â-Genus and Collapsing
    The Journal of Geometric Analysis Volume 10, Number 3, 2000 A A-Genus and Collapsing By John Lott ABSTRACT. If M is a compact spin manifold, we give relationships between the vanishing of A( M) and the possibility that M can collapse with curvature bounded below. 1. Introduction The purpose of this paper is to extend the following simple lemma. Lemma 1.I. If M is a connectedA closed Riemannian spin manifold of nonnegative sectional curvature with dim(M) > 0, then A(M) = O. Proof Let K denote the sectional curvature of M and let R denote its scalar curvature. Suppose that A(M) (: O. Let D denote the Dirac operator on M. From the Atiyah-Singer index theorem, there is a nonzero spinor field 7t on M such that D~p = 0. From Lichnerowicz's theorem, 0 = IDOl 2 dvol = IV~Pl 2 dvol + ~- I•12 dvol. (1.1) From our assumptions, R > 0. Hence V~p = 0. This implies that I~k F2 is a nonzero constant function on M and so we must also have R = 0. Then as K > 0, we must have K = 0. This implies, from the integral formula for A'(M) [14, p. 231], that A"(M) = O. [] The spin condition is necessary in Lemma 1.l, as can be seen in the case of M = CP 2k. The Ricci-analog of Lemma 1.1 is false, as can be seen in the case of M = K3. Definition 1.2. A connected closed manifold M is almost-nonnegatively-curved if for every E > 0, there is a Riemannian metric g on M such that K(M, g) diam(M, g)2 > -E.
    [Show full text]
  • Riemann Surfaces
    RIEMANN SURFACES AARON LANDESMAN CONTENTS 1. Introduction 2 2. Maps of Riemann Surfaces 4 2.1. Defining the maps 4 2.2. The multiplicity of a map 4 2.3. Ramification Loci of maps 6 2.4. Applications 6 3. Properness 9 3.1. Definition of properness 9 3.2. Basic properties of proper morphisms 9 3.3. Constancy of degree of a map 10 4. Examples of Proper Maps of Riemann Surfaces 13 5. Riemann-Hurwitz 15 5.1. Statement of Riemann-Hurwitz 15 5.2. Applications 15 6. Automorphisms of Riemann Surfaces of genus ≥ 2 18 6.1. Statement of the bound 18 6.2. Proving the bound 18 6.3. We rule out g(Y) > 1 20 6.4. We rule out g(Y) = 1 20 6.5. We rule out g(Y) = 0, n ≥ 5 20 6.6. We rule out g(Y) = 0, n = 4 20 6.7. We rule out g(C0) = 0, n = 3 20 6.8. 21 7. Automorphisms in low genus 0 and 1 22 7.1. Genus 0 22 7.2. Genus 1 22 7.3. Example in Genus 3 23 Appendix A. Proof of Riemann Hurwitz 25 Appendix B. Quotients of Riemann surfaces by automorphisms 29 References 31 1 2 AARON LANDESMAN 1. INTRODUCTION In this course, we’ll discuss the theory of Riemann surfaces. Rie- mann surfaces are a beautiful breeding ground for ideas from many areas of math. In this way they connect seemingly disjoint fields, and also allow one to use tools from different areas of math to study them.
    [Show full text]
  • Recognizing Surfaces
    RECOGNIZING SURFACES Ivo Nikolov and Alexandru I. Suciu Mathematics Department College of Arts and Sciences Northeastern University Abstract The subject of this poster is the interplay between the topology and the combinatorics of surfaces. The main problem of Topology is to classify spaces up to continuous deformations, known as homeomorphisms. Under certain conditions, topological invariants that capture qualitative and quantitative properties of spaces lead to the enumeration of homeomorphism types. Surfaces are some of the simplest, yet most interesting topological objects. The poster focuses on the main topological invariants of two-dimensional manifolds—orientability, number of boundary components, genus, and Euler characteristic—and how these invariants solve the classification problem for compact surfaces. The poster introduces a Java applet that was written in Fall, 1998 as a class project for a Topology I course. It implements an algorithm that determines the homeomorphism type of a closed surface from a combinatorial description as a polygon with edges identified in pairs. The input for the applet is a string of integers, encoding the edge identifications. The output of the applet consists of three topological invariants that completely classify the resulting surface. Topology of Surfaces Topology is the abstraction of certain geometrical ideas, such as continuity and closeness. Roughly speaking, topol- ogy is the exploration of manifolds, and of the properties that remain invariant under continuous, invertible transforma- tions, known as homeomorphisms. The basic problem is to classify manifolds according to homeomorphism type. In higher dimensions, this is an impossible task, but, in low di- mensions, it can be done. Surfaces are some of the simplest, yet most interesting topological objects.
    [Show full text]
  • Harmonic Maps from Surfaces of Arbitrary Genus Into Spheres
    Harmonic maps from surfaces of arbitrary genus into spheres Renan Assimos and J¨urgenJost∗ Max Planck Institute for Mathematics in the Sciences Leipzig, Germany We relate the existence problem of harmonic maps into S2 to the convex geometry of S2. On one hand, this allows us to construct new examples of harmonic maps of degree 0 from compact surfaces of arbitrary genus into S2. On the other hand, we produce new example of regions that do not contain closed geodesics (that is, harmonic maps from S1) but do contain images of harmonic maps from other domains. These regions can therefore not support a strictly convex function. Our construction builds upon an example of W. Kendall, and uses M. Struwe's heat flow approach for the existence of harmonic maps from surfaces. Keywords: Harmonic maps, harmonic map heat flow, maximum principle, convexity. arXiv:1910.13966v2 [math.DG] 4 Nov 2019 ∗Correspondence: [email protected], [email protected] 1 Contents 1 Introduction2 2 Preliminaries4 2.1 Harmonic maps . .4 2.2 The harmonic map flow . .6 3 The main example7 3.1 The construction of the Riemann surface . .7 3.2 The initial condition u0 ............................8 3.3 Controlling the image of u1 ......................... 10 4 Remarks and open questions 13 Bibliography 15 1 Introduction M. Emery [3] conjectured that a region in a Riemannian manifold that does not contain a closed geodesic supports a strictly convex function. W. Kendall [8] gave a counterexample to that conjecture. Having such a counterexample is important to understand the relation between convexity of domains and convexity of functions in Riemannian geometry.
    [Show full text]
  • The Arithmetic of Genus Two Curves Arxiv:1209.0439V3 [Math.AG] 6 Sep
    The arithmetic of genus two curves 1 T. SHASKA 2 Department of Mathematics, Oakland University L. BESHAJ 3 Department of Mathematics, University of Vlora. Abstract. Genus 2 curves have been an object of much mathematical interest since eighteenth century and continued interest to date. They have become an important tool in many algorithms in cryptographic applications, such as factoring large numbers, hyperelliptic curve cryp- tography, etc. Choosing genus 2 curves suitable for such applications is an important step of such algorithms. In existing algorithms often such curves are chosen using equations of moduli spaces of curves with decomposable Jacobians or Humbert surfaces. In these lectures we will cover basic properties of genus 2 curves, mod- uli spaces of (n,n)-decomposable Jacobians and Humbert surfaces, mod- ular polynomials of genus 2, Kummer surfaces, theta-functions and the arithmetic on the Jacobians of genus 2, and their applications to cryp- tography. The lectures are intended for graduate students in algebra, cryptography, and related areas. Keywords. genus two curves, moduli spaces, hyperelliptic curve cryptography, modular polynomials 1. Introduction Genus 2 curves are an important tool in many algorithms in cryptographic ap- plications, such as factoring large numbers, hyperelliptic curve cryptography, etc. arXiv:1209.0439v3 [math.AG] 6 Sep 2012 Choosing such genus 2 curves is an important step of such algorithms. Most of genus 2 cryptographic applications are based on the Kummer surface Ka;b;c;d. Choosing small a; b; c; d makes the arithmetic on the Jacobian very fast. However, only a small fraction of all choices of a; b; c; d are secure.
    [Show full text]
  • An Efficient Algorithm to Compute the Genus of Discrete Surfaces And
    Algorithm 964: An Efficient Algorithm to Compute the Genus of Discrete Surfaces and Applications to Turbulent Flows ADRIAN´ LOZANO-DURAN´ and GUILLEM BORRELL, Universidad Politecnica´ de Madrid A simple and efficient algorithm to numerically compute the genus of surfaces of three-dimensional objects using the Euler characteristic formula is presented. The algorithm applies to objects obtained by thresholding a scalar field in a structured-collocated grid and does not require any triangulation of the data. This makes the algorithm fast, memory efficient, and suitable for large datasets. Applications to the characterization of complex surfaces in turbulent flows are presented to illustrate the method. Categories and Subject Descriptors: G.4 [Mathematical Software]: Algorithm Design and Analysis; I.3 [Computational Geometry and Object Modeling]; J.2 [Physical Science and Engineering]: Physics General Terms: Algorithms Additional Key Words and Phrases: Genus, Euler characteristic, voxels, turbulence, coherent structures, turbulent/nonturbulent interface ACM Reference Format: Adrian´ Lozano-Duran´ and Guillem Borrell. 2016. An efficient algorithm to compute the genus of discrete surfaces and applications to turbulent flows. ACM Trans. Math. Softw. 42, 4, Article 34 (June 2016), 19 pages. DOI: http://dx.doi.org/10.1145/2845076 1. INTRODUCTION We present a fast and memory-efficient algorithm to numerically compute the topo- logical genus of all surfaces associated with three-dimensional objects in a discrete space. The article is aimed at the turbulence community interested in the topology of three-dimensional entities in turbulent flows such as coherent structures [del Alamo´ et al. 2006; Lozano-Duran´ et al. 2012] or turbulent/nonturbulent interfaces [da Silva et al.
    [Show full text]
  • Lesson Plans for Mobius Strip Twists
    Twists & Turns Disseminator: Marilyn Riggins [email protected] South Miami Senior High School Code: 7721 Telephone: 305-666-5871 Fax: 305-666-5437 For information concerning IMPACT II opportunities, Adapter and Disseminator grants, please contact: The Education Fund 305-892-5099, Ext. 18 e-mail: [email protected] website: www.educationfund.org 1 Table of Contents I. Goals and Objectives Sunshine State Standards. Page 3 II. Course Outline/Overview . Page 4 III. Topology Classifications . Page 5 IV. Networks . Page 7 V. Mobius Strips . Page 10 VI. Color My World . Page 13 VII. Why Knot? . Page 15 VIII. The Shape of Space Activities . Page 17 Megamath Activities IX. Classic Topological Games and Puzzles . Page 18 X. Resources . Page 19 XI. Bibliography . Page 21 XII. Adapter Application . Page 23 2 Goals and Objectives 1. Students will experience topology as the geometry of distortion. 2. Students will see topology as a deductive system of interrelated concepts. 3. Students will use patterns and inductive reasoning to summarize experimental results. 4. Students will be able to apply some topological classifications. SUNSHINE STATE STANDARDS The student describes, analyzes, and generalizes a wide variety of patterns, relations, and functions. ( MA.D.1.2, MA.D.1.3, MA.D.1.4 ) The student describes, draws, identifies, and analyzes two- and three-dimensional shapes. ( MA.C.1.3, MA.C.1.4 ) The student visualizes and illustrates ways in which shapes can be combined, subdivided, and changed. ( MA.C.2.3, MA.C. 2. 4 ) GOALS AND OBJECTIVES for Florida's Frameworks for K-12 Gifted Learners These new frameworks can used to enhance the level of rigor and intellectual challenge in classrooms of gifted learners.
    [Show full text]
  • Approximation Algorithms for Euler Genus and Related Problems
    Approximation algorithms for Euler genus and related problems Chandra Chekuri Anastasios Sidiropoulos February 3, 2014 Slides based on a presentation of Tasos Sidiropoulos Theorem (Kuratowski, 1930) A graph is planar if and only if it does not contain K5 and K3;3 as a topological minor. Theorem (Wagner, 1937) A graph is planar if and only if it does not contain K5 and K3;3 as a minor. Graphs and topology Theorem (Wagner, 1937) A graph is planar if and only if it does not contain K5 and K3;3 as a minor. Graphs and topology Theorem (Kuratowski, 1930) A graph is planar if and only if it does not contain K5 and K3;3 as a topological minor. Graphs and topology Theorem (Kuratowski, 1930) A graph is planar if and only if it does not contain K5 and K3;3 as a topological minor. Theorem (Wagner, 1937) A graph is planar if and only if it does not contain K5 and K3;3 as a minor. Minors and Topological minors Definition A graph H is a minor of G if H is obtained from G by a sequence of edge/vertex deletions and edge contractions. Definition A graph H is a topological minor of G if a subdivision of H is isomorphic to a subgraph of G. Planarity planar graph non-planar graph Planarity planar graph non-planar graph g = 0 g = 1 g = 2 g = 3 k = 1 k = 2 What about other surfaces? sphere torus double torus triple torus real projective plane Klein bottle What about other surfaces? sphere torus double torus triple torus g = 0 g = 1 g = 2 g = 3 real projective plane Klein bottle k = 1 k = 2 Genus of graphs Definition The orientable (reps.
    [Show full text]