DOCSLIB.ORG

Entropically Driven Self‐Assembly of Pear‐Shaped Nanoparticles

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Entropically Driven Self‐Assembly of Pear‐Shaped Nanoparticles UNIVERSITY The ambition to recreate highly complex and functional nanostructures found in living PRESS organisms marks one of the pillars of today‘s research in bio- and soft matter physics. Here, self-assembly has evolved into a prominent strategy in nanostructure formation and has proven to be a useful tool for many complex structures. However, it is still a challenge to design and realise particle properties such that they self-organise into a desired target configuration. One of the key design parameters is the shape of the constituent particles. This thesis focuses in particular on the shape sensitivity of liquid crystal phases by addressing the entropically driven colloidal self-assembly of tapered ellipsoids, reminiscent of „pear-shaped“ particles. Therefore, we analyse the formation of the gyroid and of the accompanying bilayer architecture, reported earlier in the so-called pear hard Gaussian overlap (PHGO) approximation, by applying various geometrical tools like Set-Voronoi tessellation and clustering algorithms. Using computational simulations, we also indicate a method to stabilise other bicontinuous structures like the diamond phase. Moreover, we investigate both computationally and theoretically FAU Forschungen, Reihe B, Medizin, Naturwissenschaft, Technik 32 (density functional theory) the influence of minor variations in shape on different pear- shaped particle systems, including the stability of the PHGO gyroid phase. We show that the formation of the gyroid is due to small non-additive properties of the PHGO potential. This phase does not form in pears with a „true“ hard pear-shaped potential. Philipp W. A. Schönhöfer Overall our results allow for a better general understanding of necessity and sufficiency of particle shape in regards to colloidal self-assembly processes. Furthermore, the pear-shaped particle system sheds light on a unique collective mechanism to generate bicontinuous phases. It suggests a new alternative pathway which might help us to solve still unknown characteristics and properties of naturally occurring Entropically driven self-assembly of gyroid-like nano- and microstructures. nanoparticles driven self-assembly of pear-shaped Entropically pear-shaped nanoparticles ISBN 978-3-96147-268-0 FAU UNIVERSITY PRESS 2012 FAU FAU UNIVERSITY PRESS 2019 FAU A. Schönhöfer W. P. Philipp W. A. Schönhöfer Entropically driven self-assembly of pear-shaped nanoparticles FAU Forschungen, Reihe B Medizin, Naturwissenschaft, Technik Band 32 Herausgeber der Reihe: Wissenschaftlicher Beirat der FAU University Press Philipp W. A. Schönhöfer Entropically driven self‐assembly of pear‐shaped nanoparticles Erlangen FAU University Press 2019 Bibliografische Information der Deutschen Nationalbibliothek: Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.d-nb.de abrufbar. Bitte zitieren als Schönhöfer, Philipp W. A. 2019. Entropically driven self‐assembly of pear‐shaped nanoparticles. FAU Forschungen, Reihe B, Medizin, Naturwissenschaft, Technik Band 32. Erlangen: FAU University Press, DOI: 10.25593/978-3-96147-269-7. Das Werk, einschließlich seiner Teile, ist urheberrechtlich geschützt. Die Rechte an allen Inhalten liegen bei ihren jeweiligen Autoren. Sie sind nutzbar unter der Creative Commons Lizenz BY. Der vollständige Inhalt des Buchs ist als PDF über den OPUS Server der Friedrich-Alexander-Universität Erlangen-Nürnberg abrufbar: https://opus4.kobv.de/opus4-fau/home Verlag und Auslieferung: FAU University Press, Universitätsstraße 4, 91054 Erlangen Druck: docupoint GmbH ISBN: 978-3-96147-268-0 (Druckausgabe) eISBN: 978-3-96147-269-7 (Online-Ausgabe) ISSN: 2198-8102 DOI: 10.25593/978-3-96147-269-7 Entropically driven self‐assembly of pear‐shaped nanoparticles Entropie‐dominierte Selbstorganisationsprozesse birnenförmiger Teilchensysteme Der Naturwissenschaftlichen Fakultät der Friedrich‐Alexander‐Universität Erlangen‐Nürnberg zur Erlangung des Doktorgrades Dr. rer. nat. vorgelegt von Philipp Wilhelm Albert Schönhöfer aus Erlangen Als Dissertation genehmigt von der Naturwissenschaftlichen Fakultät der Friedrich‐Alexander‐Universität Erlangen‐Nürnberg Tag der mündlichen Prüfung: 25. Juli 2019 Vorsitzender des Promotionsorgans: Prof. Dr. Georg Kreimer Gutachter: Prof. Dr. Klaus Mecke Gutachter: Prof. Dr. Hartmut Löwen Gutachter: Prof. Dr. Peter Harrowell Diese Arbeit ist Teil eines Cotutelle Verfahrens zwischen der Naturwissenschaftlichen Fakultät der Friedrich‐Alexander‐Universität Erlangen‐Nürnberg und der School of Engineering and Information Technology der Murdoch University, Perth blank! Abstract This thesis addresses the entropically driven colloidal self‐assembly of pear‐ shaped particle ensembles, including the formation of nanostructures based on triply periodic minimal surfaces, in particular of the Ia3d gyroid. One of the key results is that the formation of the Ia3d gyroid, reported earlier in the so‐called pear hard Gaussian overlap (PHGO) approximation and confirmed here, is due to a slight non‐additivity of that potential; this phase does not form in pears with true hard‐core potential. First, we computationally study the PHGO system and present the phase diagram of pears with an aspect ratio of 3 in terms of global density and particle shape (degree of taper), containing gyroid, isotropic, nematic and smectic phases. We confirm that it is adequate to interpret the gyroid as a warped smectic bilayer phase. The collective behaviour to arrange into interdigitated sheets with negative Gauss curvature, from which the gyroid results, is investigated through correlations of (Set‐)Voronoi cells and lo‐ cal curvature. This geometric arrangement within the bilayers suggests a fundamentally different stabilisation mechanism of the pear gyroid phase compared to those found in both lipid‐water and di‐block copolymer sys‐ tems forming the Ia3d gyroid. The PHGO model is only an approximation for hard‐core interactions, and we additionally investigate, by much slower simulations, pear‐assemblies with true hard‐core interactions (HPR). We find that HPR phase diagram only contains isotropic and nematic phases, but neither gyroid nor smectic phases. To understand this shape sensitivity more profoundly, the deple‐ tion interactions of both models are studied in two pear‐shaped colloids dis‐ solved in a hard sphere solvent. The HPR particles act as one would expect from a geometric analysis of the excluded‐volume minimisation, whereas the PHGO particles show deviations from this expectation. These differ‐ ences are attributed to the unusual angle dependency of the (non‐additive) contact function and, more so, to small overlaps induced by the approxima‐ tion. v For the PHGO model, we further demonstrate that the addition of a small concentration of hard spheres (“solvent”) drives the system towards a Pn3m diamond phase. This result is explained by the greater spatial heterogeneity of the diamond geometry compared to the gyroid where additional material is needed to relieve packing frustration. In contrast to copolymer systems, however, the solvent mostly aggregates near the diamond minimal surface, driven by the non‐additivity of the PHGO pears. At high solvent concentra‐ tions, the mixture phase separates into “inverse” micelle‐like structures with the blunt ends at the micellar centres and thin ends pointing outwards. The micelles themselves spontaneously cluster, indicative of a hierarchical self‐ assembly process for bicontinuous structures. Finally, we develop a density functional for hard solids of revolution (in‐ cluding pears) within the framework of fundamental measure theory. It is applied to low‐density ensembles of pear‐shaped particles, where we anal‐ yse their response near a hard substrate. A complex orientational ordering close to the wall is predicted, which is directly linked to the particle shape and gives insight into adsorption processes of asymmetric particles. This predicted behaviour and the differences between the PHGO and HPR model are confirmed by MC simulations. vi Zusammenfassung Diese Doktorarbeit befasst sich mit der Selbstorganisation entropisch ge‐ triebener, birnen‐förmiger Kolloidteilchen. Hierbei wird besonders auf die Bildung von Nanostrukturen eingegangen, die auf dreifach‐periodischen Minimalflächen und insbesondere auf der Ia3d Gyroid Minimalfläche fußen. Eines unserer wichtigsten Ergebnisse besteht aus der Erkenntnis, dass die spontane Bildung der Ia3d Gyroid Struktur, von welcher in früheren Studien über die pear hard Gaussian overlap (PHGO) Annäherung berichtet wurde und die hier bestätigt wurde, auf geringfügig nicht‐additive Eigenschaften dieses Potentials zurückzuführen ist; in Systemen mit Teilchen von perfekt harter Birnenform wird dieser dreifach‐periodische Flüssigkristallzustand nicht beobachtet. Zuerst untersuchen wir mit Hilfe von Computersimulationen das PHGO System, bei dem wir das Phasendiagram von Birnen‐Teilchen mit Aspektver‐ hältnis 3 präsentieren. Dieses ist in Abhängigkeit der globalen Dichte und der Teilchenform (Teilchenverjüngung) angegeben und beinhaltet sowohl gyroide, isotrope, nematische, als auch smektische Phasen. Wir bestäti‐ gen, dass die Gyroidphase als gekrümmte smektische Bilagenphase gedeutet werden kann. Das kollektive Verhalten der Birnenteilchen, sich in verzahn‐ te Schichten mit negativer Gausskrümmung, aus der der Gyroid resultiert, anzuordnen, wird erkundet, indem ein Zusammenhang zwischen den zu‐ grundeliegenden Mengen
Load more

Recommended publications
  • MINIMAL TWIN SURFACES 1. Introduction in Material Science, A
    MINIMAL TWIN SURFACES HAO CHEN Abstract. We report some minimal surfaces that can be seen as copies of a triply periodic minimal surface related by reflections in parallel planes. We call them minimal twin surfaces for the resemblance with crystal twinning. Twinning of Schwarz' Diamond and Gyroid surfaces have been observed in experiment by material scientists. In this paper, we investigate twinnings of rPD surfaces, a family of rhombohedral deformations of Schwarz' Primitive (P) and Diamond (D) surfaces, and twinnings of the Gyroid (G) surface. Small examples of rPD twins have been constructed in Fijomori and Weber (2009), where we observe non-examples near the helicoid limit. Examples of rPD twins near the catenoid limit follow from Traizet (2008). Large examples of rPD and G twins are numerically constructed in Surface Evolver. A structural study of the G twins leads to new description of the G surface in the framework of Traizet. 1. Introduction In material science, a crystal twinning refers to a symmetric coexistence of two or more crystals related by Euclidean motions. The simplest situation, namely the reflection twin, consists of two crystals related by a reflection in the boundary plane. Triply periodic minimal surfaces (TPMS) are minimal surfaces with the symmetries of crystals. They are used to model lyotropic liquid crystals and many other structures in nature. Recently, [Han et al., 2011] synthesized mesoporous crystal spheres with polyhedral hollows. A crystallographic study shows a structure of Schwarz' D (diamond) surface. Most interestingly, twin structures are observed at the boundaries of the domains; see Figure 1. In the language of crystallography, the twin boundaries are f111g planes.
    [Show full text]
  • Computation of Minimal Surfaces H
    COMPUTATION OF MINIMAL SURFACES H. Terrones To cite this version: H. Terrones. COMPUTATION OF MINIMAL SURFACES. Journal de Physique Colloques, 1990, 51 (C7), pp.C7-345-C7-362. 10.1051/jphyscol:1990735. jpa-00231134 HAL Id: jpa-00231134 https://hal.archives-ouvertes.fr/jpa-00231134 Submitted on 1 Jan 1990 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. COLLOQUE DE PHYSIQUE Colloque C7, supplkment au no23, Tome 51, ler dkcembre 1990 COMPUTATION OF MINIMAL SURFACES H. TERRONES Department of Crystallography, Birkbeck College, University of London, Malet Street, GB- ond don WC~E~HX, Great-Britain Abstract The study of minimal surfaces is related to different areas of science like Mathemat- ics, Physics, Chemistry and Biology. Therefore, it is important to make more accessi- ble concepts which in the past were used only by mathematicians. These concepts are analysed here in order to compute some minimal surfaces by solving the Weierstrass equations. A collected list of Weierstrass functions is given. Knowing these functions we can get important characteristics of a minimal surface like the metric, the unit normal vectors and the Gaussian curvature. 1 Introduction The study of Minimal Surfaces (MS) is not a new field.
    [Show full text]
  • 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
    [Show full text]
  • Introduction to Conjugate Plateau Constructions by H
    Introduction to Conjugate Plateau Constructions by H. Karcher, Bonn. Version December 2001 Abstract. We explain how geometric transformations of the solutions of carefully designed Plateau problems lead to complete, often embedded, minimal or constant mean curvature surfaces in space forms. Introduction. The purpose of this paper is, to explain to a reader who is already familiar with the the- ory of minimal surfaces another successful method to construct examples. This method is independent of the Weierstra representation which is the construction method more immediately related to the theory. This second method is called “conjugate Plateau con- struction” and we summarize it as follows: solve a Plateau problem with polygonal contour, then take its conjugate minimal surface which turns out to be bounded by planar lines of reectional symmetry (details in section 2), nally use the symmetries to extend the conjugate piece to a complete (and if possible: embedded) minimal surface. For triply periodic minimal surfaces in R3 this has been by far the simplest and richest method of construction. [KP] is an attempt to explain the method to a broader audience, beyond mathematicians. Large families of doubly and singly periodic minimal surfaces in R3 have also been obtained. By contrast, for nite total curvature minimal surfaces the Weierstra representation has been much more successful. Since conjugate minimal surfaces can also be dened in spheres and hyperbolic spaces (sec- tion 2) the method has also been used there. However in these applications the contour of the Plateau problem is not determined already by the symmetry group with which one wants to work.
    [Show full text]
  • Continuous Transformations of Cubic Minimal Surfaces
    Eur. Phys. J. B 7, 91–104 (1999) THE EUROPEAN PHYSICAL JOURNAL B EDP Sciences c Societ`a Italiana di Fisica Springer-Verlag 1999 Continuous transformations of cubic minimal surfaces A. Fogden1,a and S.T. Hyde2 1 Physical Chemistry 1, Center for Chemistry and Chemical Engineering, University of Lund, Box 124, 221 00 Lund, Sweden 2 Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Canberra ACT 0200, Australia Received: 5 March 1998 / Revised: 29 July 1998 / Accepted: 31 July 1998 Abstract. Although the primitive (P ), diamond (D)andgyroid(G) minimal surfaces form the structural basis for a multitude of self-assembling phases, such as the bicontinuous cubics, relatively little is known regarding their geometrical transformations, beyond the existence of the Bonnet isometry. Here their highest symmetry deformation modes, the rhombohedral and tetragonal distortions, are fully elucidated to provide a unified description of these simplest minimal surface families, with all quantities expressed in terms of complete elliptic integrals. The rhombohedral distortions of the gyroid are found to merge continuously with those which bridge the P and D surfaces, furnishing direct transformations between all three cubics, preserving both topology and zero mean curvature throughout. The tetragonal distortions behave analogously, offering an alternative route from the gyroid to the D surface. The cell axis ratios, surface areas and Gaussian curvature moments of all families are given, supplying the necessary geometrical input to a curvature energy description of cubic and intermediate phase stability. PACS. 61.30.-v Liquid crystals – 64.70.-p Specific phase transitions – 83.70.-f Material form 1 Introduction state is expressed as an expansion in the geometrical in- variants of the dividing surface.
    [Show full text]
  • The Classification of Singly Periodic Minimal Surfaces with Genus Zero and Scherk-Type Ends
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 359, Number 3, March 2007, Pages 965–990 S 0002-9947(06)04094-3 Article electronically published on October 16, 2006 THE CLASSIFICATION OF SINGLY PERIODIC MINIMAL SURFACES WITH GENUS ZERO AND SCHERK-TYPE ENDS JOAQU´IN PEREZ´ AND MARTIN TRAIZET Abstract. Given an integer k ≥ 2, let S(k) be the space of complete embed- ded singly periodic minimal surfaces in R3, which in the quotient have genus zero and 2k Scherk-type ends. Surfaces in S(k) can be proven to be proper, a condition under which the asymptotic geometry of the surfaces is well known. It is also known that S(2) consists of the 1-parameter family of singly periodic Scherk minimal surfaces. We prove that for each k ≥ 3, there exists a natu- ral one-to-one correspondence between S(k) and the space of convex unitary nonspecial polygons through the map which assigns to each M ∈S(k)the polygon whose edges are the flux vectors at the ends of M (a special polygon is a parallelogram with two sides of length 1 and two sides of length k − 1). As consequence, S(k) reduces to the saddle towers constructed by Karcher. 1. Introduction In 1834, Scherk [18] discovered a 1-parameter family of singly periodic, complete, properly embedded minimal surfaces in Euclidean space R3. Each of these surfaces may be seen as the desingularisation of two vertical planes: the angle θ ∈ (0,π/2] between the planes is the parameter of the family. These surfaces are now called Scherk singly periodic minimal surfaces.InthequotientR3/T by the shortest orientation-preserving translation T , these surfaces have genus zero and four ends asymptotic to flat vertical annuli (namely the quotients of vertical half-planes by T ).
    [Show full text]
  • Minimal Surfaces from Circle Patterns: Geometry from Combinatorics
    Annals of Mathematics, 164 (2006), 231–264 Minimal surfaces from circle patterns: Geometry from combinatorics ∗ ∗∗ ∗∗ By Alexander I. Bobenko , Tim Hoffmann , and Boris A. Springborn * 1. Introduction The theory of polyhedral surfaces and, more generally, the field of discrete differential geometry are presently emerging on the border of differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric shapes with a finite number of elements (polyhedra), the theory of polyhedral surfaces aims at a development of discrete equivalents of the geometric notions and methods of surface theory. The latter appears then as a limit of the refinement of the discretization. Current progress in this field is to a large extent stimulated by its relevance for computer graphics and visualization. One of the central problems of discrete differential geometry is to find proper discrete analogues of special classes of surfaces, such as minimal, con- stant mean curvature, isothermic surfaces, etc. Usually, one can suggest vari- ous discretizations with the same continuous limit which have quite different geometric properties. The goal of discrete differential geometry is to find a dis- cretization which inherits as many essential properties of the smooth geometry as possible. Our discretizations are based on quadrilateral meshes, i.e. we discretize parametrized surfaces. For the discretization of a special class of surfaces, it is natural to choose an adapted parametrization. In this paper, we investigate conformal discretizations of surfaces, i.e. discretizations in terms of circles and spheres, and introduce a new discrete model for minimal surfaces. See Figures 1 and 2.
    [Show full text]
  • Two Catenoid Punctures Are the Catenoids Themselves
    About Minimal Surfaces Discovery and physical interpretation. The question which surfaces locally minimize area led La- grange in 1760 to the minimal surface equation for graphs. By 1765 Meusnier had found that a geometric interpreta- tion of this equation is: the mean curvature of the surface vanishes. He discovered that the catenoid and the heli- coid are nonplanar examples. It took until 1835 for the next examples to appear, discovered by Scherk; his dou- bly periodic surface is a graph over the black squares of a checkerboard tesselation of the plane and his singly peri- odic surface is nowadays viewed as a desingularization of two orthogonally intersecting planes. In the following years complex analysis developed and by 1865 many examples were known through the efforts of Riemann, Weierstraß, Enneper and in particular Schwarz. Also in that period Plateau had made careful experiments with soap films. He convinced people that soap films were a perfect physical realization of minimal surfaces, and he convinced mathematicians that they should solve Plateau’s Problem, i.e. prove that every continuous injective closed curve in R3 spans a minimal surface. This problem was solved in 1932 by Douglas and independently by Rado. On the way to this solution mathematicians had learnt a lot about nonlinear elliptic partial differential equations. 1 In particular the importance of the maximum principle had become clear, it implies for example that every com- pact minimal surface is contained in the convex hull of its boundary and that boundary value problems are well posed for the minimal surface equation. On the other hand, although the Cauchy-Kowalewski the- orem allows to solve locally initial value problems with analytic data, there is no continuous dependence on the data and no hope to obtain complete immersed examples with this method.
    [Show full text]
  • A Survey on Classical Minimal Surface Theory
    A survey on classical minimal surface theory BY William H. Meeks III Joaqu´ınP´erez Preface. We present in this monograph a survey of recent spectacular successes in classical minimal surface theory. Many of these successes were covered in our survey article \The classical theory of minimal surfaces" that appeared in the Bulletin of the AMS [125]. The focus of our BAMS article was to describe the work that led up to the theorem that the plane, the helicoid, the catenoid and the one-parameter family of Riemann minimal examples 3 are the only properly embedded, minimal planar domains in R . The proof of this result depends primarily on work of Colding and Minicozzi [37, 34], Collin [38], L´opez and Ros [107], Meeks, P´erezand Ros [136] and Meeks and Rosenberg [148]. Partly due to limitations of length of our BAMS article, we omitted the discussion of many other important recent advances in the theory. The central topics missing in [125] and which appear here include the following ones: 3 1. The topological classification of minimal surfaces in R (Frohman and Meeks [63]). 2. The uniqueness of Scherk's singly-periodic minimal surfaces (Meeks and Wolf [155]). 3. The Calabi-Yau problem for minimal surfaces based on work by Nadi- rashvili [166] and Ferrer, Mart´ınand Meeks [56]. 4. Colding-Minicozzi theory for minimal surfaces of finite genus [24]. 5. The asymptotic behavior of minimal annular ends with infinite total cur- vature (Meeks and P´erez[126]). ii Preface. 6. The local removable singularity theorem for minimal laminations and its applications: quadratic decay of curvature theorem, dynamics theorem and the local picture theorem on the scale of topology (Meeks, P´erezand Ros [134]).
    [Show full text]
  • Uniqueness of the Helicoid and the Classical Theory of Minimal Surfaces
    Uniqueness of the helicoid and the classical theory of minimal surfaces William H. Meeks III∗ Joaqu´ın P´erez ,† September 4, 2006 Abstract We present here a survey of recent spectacular successes in classical minimal surface theory. We highlight this article with the theorem that the plane and the helicoid are the unique simply-connected, complete, embedded minimal surfaces in R3; the proof of this result depends primarily on work of Meeks and Rosenberg and of Colding and Minicozzi. Rather than culminating and ending the theory with this uniqueness result, significant advances continue to be made as we enter a new golden age for classical minimal surface theory. Through our telling of the story of the uniqueness of the helicoid, we hope to pass on to the general mathematical public a glimpse of the intrinsic beauty of classical minimal surface theory, and our own excitement and perspective of what is happening at this historical moment in a very classical subject. Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42. Key words and phrases: Minimal surface, minimal lamination, locally simply-connected, finite total curvature, conformal structure, harmonic function, recurrence, transience, parabolic Riemann surface, harmonic measure, universal superharmonic function, Ja- cobi function, stability, index of stability, curvature estimates, maximum principle at infinity, limit tangent cone at infinity, limit tangent plane at infinity, parking garage, blow-up on the scale of topology, minimal planar domain, isotopy class. ∗This material is based upon work for the NSF under Award No. DMS - 0405836. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the NSF.
    [Show full text]
  • A Survey on Classical Minimal Surface Theory
    A survey on classical minimal surface theory BY William H. Meeks III Joaqu´ınP´erez Preface. We present in this monograph a survey of recent spectacular successes in classical minimal surface theory. Many of these successes were covered in our survey article \The classical theory of minimal surfaces" that appeared in the Bulletin of the AMS [125]. The focus of our BAMS article was to describe the work that led up to the theorem that the plane, the helicoid, the catenoid and the one-parameter family of Riemann minimal examples 3 are the only properly embedded, minimal planar domains in R . The proof of this result depends primarily on work of Colding and Minicozzi [37, 34], Collin [38], L´opez and Ros [107], Meeks, P´erezand Ros [136] and Meeks and Rosenberg [148]. Partly due to limitations of length of our BAMS article, we omitted the discussion of many other important recent advances in the theory. The central topics missing in [125] and which appear here include the following ones: 3 1. The topological classification of minimal surfaces in R (Frohman and Meeks [63]). 2. The uniqueness of Scherk's singly-periodic minimal surfaces (Meeks and Wolf [155]). 3. The Calabi-Yau problem for minimal surfaces based on work by Nadi- rashvili [166] and Ferrer, Mart´ınand Meeks [56]. 4. Colding-Minicozzi theory for minimal surfaces of finite genus [24]. 5. The asymptotic behavior of minimal annular ends with infinite total cur- vature (Meeks and P´erez[126]). ii Preface. 6. The local removable singularity theorem for minimal laminations and its applications: quadratic decay of curvature theorem, dynamics theorem and the local picture theorem on the scale of topology (Meeks, P´erezand Ros [134]).
    [Show full text]
  • Wick Rotations of Solutions to the Minimal Surface Equation, the Zero Mean Curvature Equation and the Born–Infeld Equation
    Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:35 https://doi.org/10.1007/s12044-019-0479-7 Wick rotations of solutions to the minimal surface equation, the zero mean curvature equation and the Born–Infeld equation SHINTARO AKAMINE1,∗ and RAHUL KUMAR SINGH2 1Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8502, Japan 2School of Mathematical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneshwar, Khurda, Odisha 752 050, India *Corresponding author. E-mail: [email protected]; [email protected] MS received 11 December 2017; revised 17 January 2018; accepted 19 January 2018; published online 20 April 2019 Abstract. In this paper, we investigate relations between solutions to the minimal surface equation in Euclidean 3-space E3, the zero mean curvature equation in the Lorentz–Minkowski 3-space L3 and the Born–Infeld equation under Wick rotations. We prove that the existence conditions of real solutions and imaginary solutions after Wick rotations are written by symmetries of solutions, and reveal how real and imaginary solutions are transformed under Wick rotations. We also give a transformation method for zero mean curvature surfaces containing lightlike lines with some symmetries. As an application, we give new correspondences among some solutions to the above equations by using the non-commutativity between Wick rotations and isometries in the ambient space. Keywords. Minimal surface; zero mean curvature surface; solution to the Born–Infeld equation; Wick rotation. 2010 Mathematics Subject Classification. 53A10, 58J72, 53B30. 1. Introduction In this paper, we study the geometric relations of real analytic solutions for the following three equations: ( + 2) − + ( + 2) = , 1 fy fxx 2 fx fy fxy 1 fx fyy 0 (1) ( − 2) + + ( − 2) = , 1 gy gxx 2gx gy gxy 1 gx gyy 0 (2) ( − 2) + − ( + 2) = .
    [Show full text]