Complete Minimal Surfaces in S 3
About: This article is published in Annals of Mathematics.The article was published on 1970-11-01. It has received 702 citations till now. The article focuses on the topics: Schwarz minimal surface & Scherk surface.
Citations
More filters
01 Jul 1982
1,623 citations
TL;DR: In this article, the authors describe the systematic physical theory developed to understand the static and dynamic aspects of membrane and vesicle configurations, and the preferred shapes arise from a competition between curvature energy which derives from the bending elasticity of the membrane, geometrical constraints such as fixed surface area and fixed enclosed volume, and a signature of the bilayer aspect.
Abstract: Vesicles consisting of a bilayer membrane of amphiphilic lipid molecules are remarkably flexible surfaces that show an amazing variety of shapes of different symmetry and topology. Owing to the fluidity of the membrane, shape transitions such as budding can be induced by temperature changes or the action of optical tweezers. Thermally excited shape fluctuations are both strong and slow enough to be visible by video microscopy. Depending on the physical conditions, vesicles adhere to and unbind from each other or a substrate. This article describes the systematic physical theory developed to understand the static and dynamic aspects of membrane and vesicle configurations. The preferred shapes arise from a competition between curvature energy, which derives from the bending elasticity of the membrane, geometrical constraints such as fixed surface area and fixed enclosed volume, and a signature of the bilayer aspect. These shapes of lowest energy are arranged into phase diagrams, which separate regi...
1,555 citations
TL;DR: A new algorithm to compute stable discrete minimal surfaces bounded by a number of fixed or free boundary curves in R 3, S 3 and H 3 is presented and an algorithm that, starting from a discrete harmonic map, gives a conjugate harmonic map is presented.
Abstract: We present a new algorithm to compute stable discrete minimal surfaces bounded by a number of fixed or free boundary curves in R 3, S 3 and H 3. The algorithm makes no restr iction on the genus and can handl e singular triangulations. Additionally, we present an algorithm that, starting from a discrete harmonic map, gives a conjugate harmonic map. This can be applied to the identity map on a minimal surface to produce its conjugate minimal surface, a procedure that often yields unstable solutions to a free boundary value problem for minimal surfaces. Symmetry properties of boundary curves are respected during conjugation.
1,339 citations
Additional excerpts
...See Lawson[9] for a more detailed explanation....
[...]
TL;DR: In this paper, it was shown that a map (f>:{M,g)-+(N,h) between Riemannian manifolds which is continuous and of class L\\ is harmonic if and only if it is a critical point of the energy functional.
Abstract: (1.1) Some of the main results described in [Report] are the following (in rough terms; notations and precise references will be given below): (1) A map (f>:{M,g)-+(N,h) between Riemannian manifolds which is continuous and of class L\\ is harmonic if and only if it is a critical point of the energy functional. (2) Let (M, g) and (N, h) be compact, and <̂ 0: (M, g) -> (N, h) a map. Then ^0 can be deformed to a harmonic map with minimum energy in its homotopy class in the following cases: (a) Riem ' f t ^0; (b) dim M = 2 and n2(N) = 0. (3) Any map 0O: S m -> S can be deformed to a harmonic map provided m ^ 7. More generally, suitably restricted harmonic polynomial maps can be joined to provide harmonic maps between spheres. (4) The homotopy class of maps of degree 1 from the 2-torus T to the 2-sphere S has no harmonic representative, whatever Riemannian metrics are put on T and S. (5) If in (2) M has a smooth boundary, then various Dirichlet problems have solutions in case (a) and (b); and also when the boundary data is sufficiently small.
551 citations
TL;DR: In this article, the double diamond microdomain morphology associated to a newly discovered family of triply periodic CMC surfaces was determined by comparison of tilt series with two-dimensional image projection simulations of 3D mathematical models.
Abstract: An A/B block copolymer consists of two macromolecules bonded together. In forming an equilibrium structure, such a material may separate into distinct phases, creating domains of component A and component B. A dominant factor in the determination of the domain morphology is area-minimization of the intermaterial surface, subject to fixed volume fraction. Surfaces that satisfy this mathematical condition are said to have constant mean curvature (CMC). The geometry of such surfaces strongly influences the physical properties of the material and they have been proposed as candidates for microstructural models in a variety of physical and biological systems. We have discovered domain structures in phase-separated diblock copolymers that closely approximate periodic CMC surfaces. Transmission electron microscopy and computer simulation are used to deduce the three-dimensional micro-structure by comparison of tilt series with two-dimensional image projection simulations of 3-D mathematical models. Three structures are discussed here, the first of which is the double diamond microdomain morphology associated to a newly discovered family of triply periodic CMC surfaces1. Second, a doubly periodic 90° twist boundary between lamellar microdomains, corresponding to a classically known surface (called Scherk's first surface), is described. Finally, we show a lamellar-catenoid microstructure that appears during rearrangement of a lamellar morphology in thin films and is apparently related to a new family of periodic surfaces.
428 citations