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Embedded Minimal Surfaces, Exotic Spheres, and Manifolds with Positive Ricci Curvature
Annals of Mathematics Embedded Minimal Surfaces, Exotic Spheres, and Manifolds with Positive Ricci Curvature Author(s): William Meeks III, Leon Simon and Shing-Tung Yau Source: Annals of Mathematics, Second Series, Vol. 116, No. 3 (Nov., 1982), pp. 621-659 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/2007026 Accessed: 06-02-2018 14:54 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics This content downloaded from 129.93.180.106 on Tue, 06 Feb 2018 14:54:09 UTC All use subject to http://about.jstor.org/terms Annals of Mathematics, 116 (1982), 621-659 Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature By WILLIAM MEEKS III, LEON SIMON and SHING-TUNG YAU Let N be a three dimensional Riemannian manifold. Let E be a closed embedded surface in N. Then it is a question of basic interest to see whether one can deform : in its isotopy class to some "canonical" embedded surface. From the point of view of geometry, a natural "canonical" surface will be the extremal surface of some functional defined on the space of embedded surfaces. -
Scalable Synthesis of Gyroid-Inspired Freestanding Three-Dimensional Graphene Architectures† Cite This: DOI: 10.1039/C9na00358d Adrian E
Nanoscale Advances View Article Online COMMUNICATION View Journal Scalable synthesis of gyroid-inspired freestanding three-dimensional graphene architectures† Cite this: DOI: 10.1039/c9na00358d Adrian E. Garcia,a Chen Santillan Wang, a Robert N. Sanderson,b Kyle M. McDevitt,a c ac a d Received 5th June 2019 Yunfei Zhang, Lorenzo Valdevit, Daniel R. Mumm, Ali Mohraz a Accepted 16th September 2019 and Regina Ragan * DOI: 10.1039/c9na00358d rsc.li/nanoscale-advances Three-dimensional porous architectures of graphene are desirable for Introduction energy storage, catalysis, and sensing applications. Yet it has proven challenging to devise scalable methods capable of producing co- Porous 3D architectures composed of graphene lms can continuous architectures and well-defined, uniform pore and liga- improve performance of carbon-based scaffolds in applications Creative Commons Attribution-NonCommercial 3.0 Unported Licence. ment sizes at length scales relevant to applications. This is further such as electrochemical energy storage,1–4 catalysis,5 sensing,6 complicated by processing temperatures necessary for high quality and tissue engineering.7,8 Graphene's remarkably high elec- graphene. Here, bicontinuous interfacially jammed emulsion gels trical9 and thermal conductivities,10 mechanical strength,11–13 (bijels) are formed and processed into sacrificial porous Ni scaffolds for large specic surface area,14 and chemical stability15 have led to chemical vapor deposition to produce freestanding three-dimensional numerous explorations of this multifunctional material due to turbostratic graphene (bi-3DG) monoliths with high specificsurface its promise to impact multiple elds. While these specic area. Scanning electron microscopy (SEM) images show that the bi- applications typically require porous 3D architectures, gra- 3DG monoliths inherit the unique microstructural characteristics of phene growth has been most heavily investigated on 2D their bijel parents. -
Optical Properties of Gyroid Structured Materials: from Photonic Crystals to Metamaterials
REVIEW ARTICLE Optical properties of gyroid structured materials: from photonic crystals to metamaterials James A. Dolan1;2;3, Bodo D. Wilts2;4, Silvia Vignolini5, Jeremy J. Baumberg3, Ullrich Steiner2;4 and Timothy D. Wilkinson1 1 Department of Engineering, University of Cambridge, JJ Thomson Avenue, CB3 0FA, Cambridge, United Kingdom 2 Cavendish Laboratory, Department of Physics, University of Cambridge, JJ Thomson Avenue, CB3 0HE, Cambridge, United Kingdom 3 NanoPhotonics Centre, Cavendish Laboratory, Department of Physics, University of Cambridge, JJ Thomson Avenue, CB3 0HE, Cambridge, United Kingdom 4 Adolphe Merkle Institute, University of Fribourg, Chemin des Verdiers 4, CH-1700 Fribourg, Switzerland 5 Department of Chemistry, University of Cambridge, Lensfield Road, CB2 1EW, Cambridge, United Kingdom E-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected] Abstract. The gyroid is a continuous and triply periodic cubic morphology which possesses a constant mean curvature surface across a range of volumetric fill fractions. Found in a variety of natural and synthetic systems which form through self-assembly, from butterfly wing scales to block copolymers, the gyroid also exhibits an inherent chirality not observed in any other similar morphologies. These unique geometrical properties impart to gyroid structured materials a host of interesting optical properties. Depending on the length scale on which the constituent materials are organised, these properties arise from starkly different physical mechanisms (such as a complete photonic band gap for photonic crystals and a greatly depressed plasma frequency for optical metamaterials). This article reviews the theoretical predictions and experimental observations of the optical properties of two fundamental classes of gyroid structured materials: photonic crystals (wavelength scale) and metamaterials (sub- wavelength scale). -
Surfaces Minimales : Theorie´ Variationnelle Et Applications
SURFACES MINIMALES : THEORIE´ VARIATIONNELLE ET APPLICATIONS FERNANDO CODA´ MARQUES - IMPA (NOTES BY RAFAEL MONTEZUMA) Contents 1. Introduction 1 2. First variation formula 1 3. Examples 4 4. Maximum principle 5 5. Calibration: area-minimizing surfaces 6 6. Second Variation Formula 8 7. Monotonicity Formula 12 8. Bernstein Theorem 16 9. The Stability Condition 18 10. Simons' Equation 29 11. Schoen-Simon-Yau Theorem 33 12. Pointwise curvature estimates 38 13. Plateau problem 43 14. Harmonic maps 51 15. Positive Mass Theorem 52 16. Harmonic Maps - Part 2 59 17. Ricci Flow and Poincar´eConjecture 60 18. The proof of Willmore Conjecture 65 1. Introduction 2. First variation formula Let (M n; g) be a Riemannian manifold and Σk ⊂ M be a submanifold. We use the notation jΣj for the volume of Σ. Consider (x1; : : : ; xk) local coordinates on Σ and let @ @ gij(x) = g ; ; for 1 ≤ i; j ≤ k; @xi @xj 1 2 FERNANDO CODA´ MARQUES - IMPA (NOTES BY RAFAEL MONTEZUMA) be the components of gjΣ. The volume element of Σ is defined to be the p differential k-form dΣ = det gij(x). The volume of Σ is given by Z Vol(Σ) = dΣ: Σ Consider the variation of Σ given by a smooth map F :Σ × (−"; ") ! M. Use Ft(x) = F (x; t) and Σt = Ft(Σ). In this section we are interested in the first derivative of Vol(Σt). Definition 2.1 (Divergence). Let X be a arbitrary vector field on Σk ⊂ M. We define its divergence as k X (1) divΣX(p) = hrei X; eii; i=1 where fe1; : : : ; ekg ⊂ TpΣ is an orthonormal basis and r is the Levi-Civita connection with respect to the Riemannian metric g. -
Generating Carbon Schwarzites Via Zeolite-Templating
Generating carbon schwarzites via zeolite-templating Efrem Brauna, Yongjin Leeb,c, Seyed Mohamad Moosavib, Senja Barthelb, Rocio Mercadod, Igor A. Baburine, Davide M. Proserpiof,g, and Berend Smita,b,1 aDepartment of Chemical and Biomolecular Engineering, University of California, Berkeley, CA 94720; bInstitut des Sciences et Ingenierie´ Chimiques (ISIC), Valais, Ecole´ Polytechnique Fed´ erale´ de Lausanne (EPFL), CH-1951 Sion, Switzerland; cSchool of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China; dDepartment of Chemistry, University of California, Berkeley, CA 94720; eTheoretische Chemie, Technische Universitat¨ Dresden, 01062 Dresden, Germany; fDipartimento di Chimica, Universita` degli Studi di Milano, 20133 Milan, Italy; and gSamara Center for Theoretical Materials Science (SCTMS), Samara State Technical University, Samara 443100, Russia Edited by Monica Olvera de la Cruz, Northwestern University, Evanston, IL, and approved July 9, 2018 (received for review March 25, 2018) Zeolite-templated carbons (ZTCs) comprise a relatively recent ZTC can be seen as a schwarzite (24–28). Indeed, the experi- material class synthesized via the chemical vapor deposition of mental properties of ZTCs are exactly those which have been a carbon-containing precursor on a zeolite template, followed predicted for schwarzites, and hence ZTCs are considered as by the removal of the template. We have developed a theoret- promising materials for the same applications (21, 22). As we ical framework to generate a ZTC model from any given zeolite will show, the similarity between ZTCs and schwarzites is strik- structure, which we show can successfully predict the structure ing, and we explore this similarity to establish that the theory of known ZTCs. We use our method to generate a library of of schwarzites/TPMSs is a useful concept to understand ZTCs. -
Minimal Surfaces Saint Michael’S College
MINIMAL SURFACES SAINT MICHAEL’S COLLEGE Maria Leuci Eric Parziale Mike Thompson Overview What is a minimal surface? Surface Area & Curvature History Mathematics Examples & Applications http://www.bugman123.com/MinimalSurfaces/Chen-Gackstatter-large.jpg What is a Minimal Surface? A surface with mean curvature of zero at all points Bounded VS Infinite A plane is the most trivial minimal http://commons.wikimedia.org/wiki/File:Costa's_Minimal_Surface.png surface Minimal Surface Area Cube side length 2 Volume enclosed= 8 Surface Area = 24 Sphere Volume enclosed = 8 r=1.24 Surface Area = 19.32 http://1.bp.blogspot.com/- fHajrxa3gE4/TdFRVtNB5XI/AAAAAAAAAAo/AAdIrxWhG7Y/s160 0/sphere+copy.jpg Curvature ~ rate of change More Curvature dT κ = ds Less Curvature History Joseph-Louis Lagrange first brought forward the idea in 1768 Monge (1776) discovered mean curvature must equal zero Leonhard Euler in 1774 and Jean Baptiste Meusiner in 1776 used Lagrange’s equation to find the first non- trivial minimal surface, the catenoid Jean Baptiste Meusiner in 1776 discovered the helicoid Later surfaces were discovered by mathematicians in the mid nineteenth century Soap Bubbles and Minimal Surfaces Principle of Least Energy A minimal surface is formed between the two boundaries. The sphere is not a minimal surface. http://arxiv.org/pdf/0711.3256.pdf Principal Curvatures Measures the amount that surfaces bend at a certain point Principal curvatures give the direction of the plane with the maximal and minimal curvatures http://upload.wikimedia.org/wi -
Schwarz' P and D Surfaces Are Stable
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Differential Geometry and its Applications 2 (1992) 179-195 179 North-Holland Schwarz’ P and D surfaces are stable Marty Ross Department of Mathematics, Rice University, Houston, TX 77251, U.S.A. Communicated by F. J. Almgren, Jr. Received 30 July 1991 Ross, M., Schwarz’ P and D surfaces are stable, Diff. Geom. Appl. 2 (1992) 179-195. Abstract: The triply-periodic minimal P surface of Schwarz is investigated. It is shown that this surface is stable under periodic volume-preserving variations. As a consequence, the conjugate D surface is also stable. Other notions of stability are discussed. Keywords: Minimal surface, stable, triply-periodic, superfluous. MS classijication: 53A. 1. Introduction The existence of naturally occuring periodic phenomena is of course well known, Recently there have been suggestions that certain periodic behavior, including surface interfaces, might be suitably modelled by triply-periodic minimal and constant mean curvature surfaces (see [18, p. 2401, [l], [2] and th e references cited there). This is not unreasonable as such surfaces are critical points for a simple functional, the area functional. However, a naturally occuring surface ipso facto exhibits some form of stability, and thus we are led to consider as well the second variation of any candidate surface. Given an oriented immersed surface M = z(S) 5 Iw3, we can consider a smooth variation Mt = z(S,t) of M, and the resulting effect upon the area, A(t). M = MO is minimal if A’(0) = 0 for every compactly supported variation which leaves dM fixed. -
MINIMAL SURFACES THAT GENERALIZE the ENNEPER's SURFACE 1. Introduction
Novi Sad J. Math. Vol. 40, No. 2, 2010, 17-22 MINIMAL SURFACES THAT GENERALIZE THE ENNEPER'S SURFACE Dan Dumitru1 Abstract. In this article we give the construction of some minimal surfaces in the Euclidean spaces starting from the Enneper's surface. AMS Mathematics Subject Classi¯cation (2000): 53A05 Key words and phrases: Enneper's surface, minimal surface, curvature 1. Introduction We start with the well-known Enneper's surface which is de¯ned in the 2 3 u3 3-dimensional Cartesian by the application x : R ! R ; x(u; v) = (u ¡ 3 + 2 v3 2 2 2 uv ; v¡ 3 +vu ; u ¡v ): The point (u; v) is mapped to (f(u; v); g(u; v); h(u; v)), which is a point on Enneper's surface [2]. This surface has self-intersections and it is one of the ¯rst examples of minimal surface. The fact that Enneper's surface is minimal is a consequence of the following theorem: u3 2 Theorem 1.1. [1] Consider the Enneper's surface x(u; v) = (u ¡ 3 + uv ; v ¡ v3 2 2 2 3 + vu ; u ¡ v ): Then: a) The coe±cients of the ¯rst fundamental form are (1.1) E = G = (1 + u2 + v2)2;F = 0: b) The coe±cients of the second fundamental form are (1.2) e = 2; g = ¡2; f = 0: c) The principal curvatures are 2 2 (1.3) k = ; k = ¡ : 1 (1 + u2 + v2)2 2 (1 + u2 + v2)2 Eg+eG¡2F f It follows that the mean curvature H = EG¡F 2 is zero, which means the Enneper's surface is minimal. -
Minimal Surfaces, a Study
Alma Mater Studiorum · Università di Bologna SCUOLA DI SCIENZE Corso di Laurea in Matematica MINIMAL SURFACES, A STUDY Tesi di Laurea in Geometria Differenziale Relatore: Presentata da: Dott.ssa CLAUDIA MAGNAPERA ALESSIA CATTABRIGA VI Sessione Anno Accademico 2016/17 Abstract Minimal surfaces are of great interest in various fields of mathematics, and yet have lots of application in architecture and biology, for instance. It is possible to list different equivalent definitions for such surfaces, which correspond to different approaches. In the following thesis we will go through some of them, concerning: having mean cur- vature zero, being the solution of Lagrange’s partial differential equation, having the harmonicity property, being the critical points of the area functional, being locally the least-area surface with a given boundary and proving the existence of a solution of the Plateau’s problem. Le superfici minime, sono di grande interesse in vari campi della matematica, e parec- chie sono le applicazioni in architettura e in biologia, ad esempio. È possibile elencare diverse definizioni equivalenti per tali superfici, che corrispondono ad altrettanti approcci. Nella seguente tesi ne affronteremo alcuni, riguardanti: la curvatura media, l’equazione differenziale parziale di Lagrange, la proprietà di una funzione di essere armonica, i punti critici del funzionale di area, le superfici di area minima con bordo fissato e la soluzione del problema di Plateau. 2 Introduction The aim of this thesis is to go through the classical minimal surface theory, in this specific case to state six equivalent definitions of minimal surface and to prove the equiv- alence between them. -
Deformations of the Gyroid and Lidinoid Minimal Surfaces
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES ADAM G. WEYHAUPT Abstract. The gyroid and Lidinoid are triply periodic minimal surfaces of genus 3 embed- ded in R3 that contain no straight lines or planar symmetry curves. They are the unique embedded members of the associate families of the Schwarz P and H surfaces. In this paper, we prove the existence of two 1-parameter families of embedded triply periodic minimal surfaces of genus 3 that contain the gyroid and a single 1-parameter family that contains the Lidinoid. We accomplish this by using the flat structures induced by the holomorphic 1 1-forms Gdh, G dh, and dh. An explicit parametrization of the gyroid using theta functions enables us to find a curve of solutions in a two-dimensional moduli space of flat structures by means of an intermediate value argument. Contents 1. Introduction 2 2. Preliminaries 3 2.1. Parametrizing minimal surfaces 3 2.2. Cone metrics 5 2.3. Conformal quotients of triply periodic minimal surfaces 5 3. Parametrization of the gyroid and description of the periods 7 3.1. The P Surface and tP deformation 7 3.2. The period problem for the P surface 11 3.3. The gyroid 14 4. Proof of main theorem 16 4.1. Sketch of the proof 16 4.2. Horizontal and vertical moduli spaces for the tG family 17 4.3. Proof of the tG family 20 5. The rG and rL families 25 5.1. Description of the Lidinoid 25 5.2. Moduli spaces for the rL family 29 5.3. -
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. -
Minimal Surfaces in Germs of Hyperbolic 3–Manifolds 1 Introduction
ISSN 1464-8997 (on line) 1464-8989 (printed) 69 eometry & opology onographs G T M Volume 7: Proceedings of the Casson Fest Pages 69–100 Minimal surfaces in germs of hyperbolic 3–manifolds Clifford Henry Taubes Abstract This article introduces a universal moduli space for the set whose archetypal element is a pair that consists of a metric and second fundamental form from a compact, oriented, positive genus minimal sur- face in some hyperbolic 3–manifold. This moduli space is a smooth, finite dimensional manifold with canonical maps to both the cotangent bundle of the Teichm¨uller space and the space of SO3(C) representations for the given genus surface. These two maps embed the universal moduli space as a Lagrangian submanifold in the product of the latter two spaces. AMS Classification 53C42, 53A10; 53D30 Keywords Hyperbolic 3–manifold, minimal surface 1 Introduction Immersed, compact, minimal surfaces are now known to appear in every com- pact, hyperbolic 3–manifold. Indeed, Schoen and Yau [14] proved that a Haken hyperbolic manifold has at least one stable minimal surface. More recently, Pitts and Rubinstein [11, 10, 12], (see also [3]) proved that all compact, hyperbolic 3–manifolds have at least one unstable, immersed, minimal surface. Other au- thors, for example Freedman, Hass and Scott [8] and Hass and Scott [9] also have foundational papers on this subject. The ubiquity of minimal surfaces in hyperbolic 3–manifolds motivates the introduction and study of a universal moduli space for the set whose archetypal element is a pair that consists of a metric and second fundamental form from a compact, oriented, positive genus minimal surface in some hyperbolic 3–manifold.