On Algebraic Minimal Surfaces

Total Page:16

File Type:pdf, Size:1020Kb

On Algebraic Minimal Surfaces On algebraic minimal surfaces Boris Odehnal December 21, 2016 Abstract 1 Introduction Minimal surfaces have been studied from many different points of view. Boundary We give an overiew on various constructions value problems, uniqueness results, stabil- of algebraic minimal surfaces in Euclidean ity, and topological problems related to three-space. Especially low degree exam- minimal surfaces have been and are stil top- ples shall be studied. For that purpose, we ics for investigations. There are only a use the different representations given by few results on algebraic minimal surfaces. Weierstraß including the so-called Björ- Most of them were published in the second ling formula. An old result by Lie dealing half of the nine-teenth century, i.e., more with the evolutes of space curves can also or less in the beginning of modern differen- be used to construct minimal surfaces with tial geometry. Only a few publications by rational parametrizations. We describe a Lie [30] and Weierstraß [50] give gen- one-parameter family of rational minimal eral results on the generation and the prop- surfaces which touch orthogonal hyperbolic erties of algebraic minimal surfaces. This paraboloids along their curves of constant may be due to the fact that computer al- Gaussian curvature. Furthermore, we find gebra systems were not available and clas- a new class of algebraic and even rationally sical algebraic geometry gained less atten- parametrizable minimal surfaces and call tion at that time. Many of the compu- them cycloidal minimal surfaces. tations are hard work even nowadays and synthetic reasoning is somewhat uncertain. Keywords: minimal surface, algebraic Besides some general work on minimal sur- surface, rational parametrization, poly- faces like [5, 8, 43, 44], there were some iso- nomial parametrization, meromorphic lated results on algebraic minimal surfaces function, isotropic curve, Weierstraß- concerned with special tasks: minimal sur- representation, Björling formula, evolute of faces on certain scrolls [22, 35, 47, 49, 53], a spacecurve, curve of constant slope. minimal surfaces related to congruences of lines [25, 28, 34, 38] minimal surfaces with MSC 2010: 53A10, 53A99, 53C42, 49Q05, a given geodesic [23], minimal surfaces of a 14J26, 14Mxx. certain degree, class, or genus (whether real 1 or not) [1, 10, 11, 19, 20, 21, 31, 41, 42, 48], Theorem 1.2. Let a minimal surface be minimal surfaces touching surfaces along tangent to a cylinder . If is algebraic,M Z M special curves [22], minimal surfaces show- then the orthogonal cross-section c of is ing special symmetries [14, 15, 16, 17], or the evolute of an algebraic curve. If c isZ the minimal surface which allow isometries to evolute of a transcendental curve, then M special classes of surfaces [4, 6, 18, 52]. is also transcendental. The famous algebraic minimal surface by Rib- Enneper which is of degree 9 and class However, according to a theorem by aucour Enneper 6 attracted intensive investigation. Con- , ’s surface, like many sequently, researchers have found different other minimal surfaces, appears as the cen- generations of this surface: as the envelope tral envelope of isotropic congruences of of the planes of symmetry of all points on lines, see [25, 28, 34, 38, 45]. the pair of focal parabolas Among the real algebraic minimal sur- faces, Enneper’s surface has lowest pos- 4 2 2 1 p1(u)=( 3 u, 0, 3 u 3 ), sible degree 9. But there are algebraic min- 4 1 −2 2 p2(v)=(0, v, v ) imal surfaces that can be found in [12, 13, 3 3 − 3 21, 30] which are of degree 3 and 4 having or as the unique minimal surface (22) the equations through the rational curve 1 : (x iy)4 + 3(x2 + y2 + z2)=0 γ(t)= t t3, t2, 0 G − − 3 and having γ’s normals for its surface normals. Since γ is planar, the surface normals of the : 2(x iy)3 6i(x iy)z 3(x+iy)=0 uniquely defined minimal surface form a de- L − − − − velopable surface (to be precise, a plane), with respect to a properly chosen Cartesian and thus, γ is a planar geodesic on En- coordinate system. The surfaces and G L neper’s minimal surface. The plane of γ is have no real equation (polynomial equation a plane of symmetry of Enneper’s surface. with real coefficients exclusively) and do not This is a manifestation of a more general carry a single real point. result by Henneberg, see [21, 24, 30, 33]: is usually called Geiser’s surface and G Lie Geiser L Theorem 1.1. A minimal surface car- is named after . ’s minimal sur- ries a planar and not straight curveMc as a face is a minimal surface of revolution with geodesic. If is algebraic, then the invo- an isotropic axis. Obviously, it is of degree M lutes of c have to be algebraic or c is the 4 and some computation tells us that the equation of its dual surface ⋆, i.e., the sur- evolute of a planar algebraic curve. G face of its tangent planes has the equation We shall make use of this fact later in ⋆ 2 4 2 2 2 3 Sec. 7 when we construct cycloidal minimal : 9w0(w1 iw2) (w1 +w2 +w3) =0 surfaces. G − − A further result due to Henneberg (see which is, therefore, of degree 6, and thus, [21, 24, 30, 33]) is the following is of class 6. G 2 Whereas Lie’s surface is of degree 3 and the parametrization also of class 3 since the implicit equation of the dual surface ⋆ reads c3uS3v 3cuSv L − f(u, v)= s3uS3v + 3suSv (1) ⋆ 2 2 2 3 : 27w0(w2+iw1) +9i(w1+w2)w3 4iw3 =0. 3c uC v L − 2 2 Geiser’s surface meets the ideal plane in is an example for that, since the implicit the same ideal line as Lie’s surface does. equation of its dual surface equals The ideal line x iy =0 is a 4-fold line on 2 2 2 − u0(u1 + u2) + and a 3-fold line on . It is remarkable (2) G L +u (u2 u2)(3u2 +3u2 +2u2)=0. that complex (non-real) algebraic minimal 3 1 − 2 1 2 3 surfaces have been undergoing detailed in- The algebraic degree of Henneberg’s sur- vestigations, see, e.g., [1, 10, 12, 13, 48]. face equals 15. Enneper’s surface is the In [30], Lie gives a result dealing with the only known example of a minimal surface ideal curves of algebraic minimal surfaces: where the degree and class sum up to 15: the degree equals 9 (cf. (23)), the class Theorem 1.3. The intersection of an al- equals 6 (cf. (24)). gebraic minimal surface with the plane at Lie gives also results on the class of an infinity consists of finitely many lines. algebraic minimal surface: Some of the ideal lines on a minimal Theorem 1.5. The class of an orientable surface may have higher multiplicities and algebraic minimal surface is alwas even. pairs of complex conjugate lines can also Henneberg occcur. ’s surface is of class 5 and For the coordinatization of ideal points non-orientable. The rational minimal and lines we refer to Sec. 2. Möbius strip given in [35] is of class 15. The results on degrees, ranks, and classes In Sec. 2, we introduce coordinates and of real algebraic minimals surfaces differ define all necessary abbreviations. Then, from the results on complex algebraic min- the different parametrization techniques for imal surfaces. For real algebraic minimal minimal surfaces are collected. Proofs for surfaces we have (see [30]) these can be found in most of the standard monographs on minimal surfaces or differ- Theorem 1.4. The sum of the degree and ential geometry such as [2, 33, 46]. Sec. 3 class of a real algebraic minimal surface is is dedicated to Enneper’s surface and its at least 15. natural generalizations. In Sec. 4, Bour’s minimal surfaces gain attention. We show The two aforementioned examples of different ways to find these minimal sur- complex minimal surfaces obviously show a faces and give estimates on the algebraic different behaviour. degrees of these surfaces. Then, in Sec. It is well-kown (cf. [30, 33]) that 5 is the 5, Richmond’s surface appears as one in lowest possible class of a real algebraic min- a one-parameter family. Sec. 6 gives ad- imal surface. Henneberg’s surface with ditional and apparently new results on a 3 well-known kind of minimal surface tangent 816, and 969 terms provided that no special to a hyperbolic paraboloid. Sec. 7 deals coordinate system is chosen and that the with an apparently new class of minimal equations are expanded in full length. surfaces. The fact that cycloids (cycloidal curves with cusps) have rational normals and are algebraic as well as their evolutes 2 Prerequisites and involutes are (see [32, 51, 55, 56]), al- lows us to construct a family of algebraic Since we are dealing with minimal surfaces minimal surfaces that admit even rational in the Euclidean three-space, Cartesian co- parametrizations. We debunk their rela- ordinates (x, y, z) are sufficient. Vectors tions to curves of constant slope on quadrics and matrices are written in bold characters. of revolution. The canonical innerproduct of two vectors u, v R3 is denoted by u, v . The Eu- The reasons for the interest in algebraic ∈ h i clidean length v of a vector v is then and, especially in rational minimal sur- k k faces are manifold: Rational parametriza- given by v = v, v . The induced crossproductk k of two vectorsh i u, v R3 is the tions can be converted into a geometrically p ∈ vector u v R3.
Recommended publications
  • 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.
    [Show full text]
  • Riemannian Submanifolds: a Survey
    RIEMANNIAN SUBMANIFOLDS: A SURVEY BANG-YEN CHEN Contents Chapter 1. Introduction .............................. ...................6 Chapter 2. Nash’s embedding theorem and some related results .........9 2.1. Cartan-Janet’s theorem .......................... ...............10 2.2. Nash’s embedding theorem ......................... .............11 2.3. Isometric immersions with the smallest possible codimension . 8 2.4. Isometric immersions with prescribed Gaussian or Gauss-Kronecker curvature .......................................... ..................12 2.5. Isometric immersions with prescribed mean curvature. ...........13 Chapter 3. Fundamental theorems, basic notions and results ...........14 3.1. Fundamental equations ........................... ..............14 3.2. Fundamental theorems ............................ ..............15 3.3. Basic notions ................................... ................16 3.4. A general inequality ............................. ...............17 3.5. Product immersions .............................. .............. 19 3.6. A relationship between k-Ricci tensor and shape operator . 20 3.7. Completeness of curvature surfaces . ..............22 Chapter 4. Rigidity and reduction theorems . ..............24 4.1. Rigidity ....................................... .................24 4.2. A reduction theorem .............................. ..............25 Chapter 5. Minimal submanifolds ....................... ...............26 arXiv:1307.1875v1 [math.DG] 7 Jul 2013 5.1. First and second variational formulas
    [Show full text]
  • Minimal Surfaces As Isotropic Curves in C3: Associated Minimal Surfaces and the BjOrling’S Problem by Kai-Wing Fung
    Minimal Surfaces as Isotropic Curves in C3: Associated minimal surfaces and the BjÄorling's problem by Kai-Wing Fung Submitted to the Department of Mathematics on December 1, 2004, in partial ful¯llment of the requirements for the degree of Bachelor of Science in Mathematics Abstract In this paper, we introduce minimal surfaces as isotropic curves in C3. Given such a isotropic curve, we can de¯ne the adjoint surface and the family of associate minimal surfaces to a minimal surface that is the real part of the isotropic curve. We study the behavior of asymptotic lines and curvature lines in a family of associate surfaces, speci¯cally the asymptotic lines of a minimal surface are the curvature lines of its adjoint surface, and vice versa. In the second part of the paper, we describe the BjÄorling's problem. Given a real- analytic curve and a real-analytic vector ¯eld along the curve, BjÄorling's problem is to ¯nd a minimal surface that includes the curve such that its unit normal ¯eld coincides with the given vector ¯eld. We shows that the BjÄorling's problem always has a unique solution. We will use some examples to demonstrate how to construct minimal surface using the results from the BjÄorling's problem. Some symmetry properties can be derived from the solution to the BjÄorling's problem. For example, straight lines are lines of rotational symmetry, and planar geodesics are lines of mirror symmetry in a minimal surface. These results are useful in solving the Schwarzian chain problem, which is to ¯nd a minimal surface that spanned into a frame that consists of ¯nitely many straight lines and planes.
    [Show full text]
  • 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.
    [Show full text]
  • Complete Embedded Minimal Surfaces of Finite Total Curvature
    Complete Embedded Minimal Surfaces of Finite Total Curvature David Hoffman∗ Hermann Karcher† MSRI Math. Institut 1000 Centennial Drive Universitat Bonn Berkeley, CA 94720 D-5300 Bonn, Germany March 16, 1995 Contents 1 Introduction 2 2 Basic theory and the global Weierstrass representation 4 2.1 Finite total curvature .............................. 14 2.2 The example of Chen-Gackstatter ........................ 16 2.3 Embeddedness and finite total curvature: necessary conditions ....... 20 2.3.1 Flux .................................... 24 2.3.2 Torque ................................... 27 2.3.3 The Halfspace Theorem ......................... 29 2.4 Summary of the necessary conditions for existence of complete embedded minimal surfaces with finite total curvature .................. 31 3 Examples with restricted topology: existence and rigidity 32 3.1 Complete embedded minimal surfaces of finite total curvature and genus zero: arXiv:math/9508213v1 [math.DG] 9 Aug 1995 the Lopez-Ros theorem .............................. 37 4 Construction of the deformation family with three ends 43 4.1 Hidden conformal symmetries .......................... 45 4.2 The birdcage model ............................... 47 4.3 Meromorphic functions constructed by conformal mappings ......... 50 4.3.1 The function T and its relationship to u................ 51 ∗Supported by research grant DE-FG02-86ER25015 of the Applied Mathematical Science subprogram of the Office of Energy Research, U.S. Department of Energy, and National Science Foundation, Division of Mathematical Sciences research grants DMS-9101903 and DMS-9312087 at the University of Massachusetts. †Partially supported by Sonderforschungsbereich SFB256 at Bonn. 1 4.4 The function z and the equation for the Riemann surface in terms of z and u 52 4.5 The Weierstrass data ............................... 55 4.6 The logarithmic growth rates .........................
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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 .
    [Show full text]
  • 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.
    [Show full text]
  • Minimal Surface System in Euclidean Four-Space 3
    MINIMAL SURFACE SYSTEM IN EUCLIDEAN FOUR-SPACE HOJOO LEE An homage to Robert Osserman’s book A Survey of Minimal Surfaces 1. MAIN RESULTS Osserman [31, 32, 33] initiated the study of the minimal surface system in arbitrary codimensions. In 1977, Lawson and Osserman [24] found fascinating counterexamples to the existence, uniqueness and regularity of solutions to the minimal surface system. As a remarkable nonlinear extension of Riemann’s removable singularity theorem, Bers’ Theorem [1, 3, 11, 33] reveals that an isolated singularity of a single valued solution to the minimal surface equation in R3 is removable. However, as in Example 2.3, the minimal surface system in R4 can have solutions with isolated non-removable singularities. Extending Bernstein’s Theorem that the only entire minimal graphs in R3 are planes, Osserman established that any entire two dimensional minimal graph in R4 should be degenerate, in the sense that its generalized Gauss map (Definition 3.1) lies on a hyper- plane of the complex projective space CP3. Landsberg [21] investigated the systems of the first order whose solutions induce minimal varieties. The classical Cauchy–Riemann equations f , f g , g solves the minimal surface system of the second order x y = y − x 2 2 2 2 ¡ 0 1¢ ¡fy gy ¢ fxx 2 fx fy gx gy fxy 1 fx gx fyy , = + 2 + 2 − + + + 2+ 2 (0 1 fy gy gxx 2 fx fy gx gy gxy 1 fx gx gyy , = ¡ + + ¢ − ¡ + ¢ +¡ + + ¢ which geometrically¡ means that¢ the two¡ dimensional¢ graph¡ of height¢ functions f (x, y) and g(x, y) is a minimal surface in R4.
    [Show full text]
  • 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.
    [Show full text]