General relativity and the Einstein equations. DIFFERENTIAL EQUATION DEFINITION •A surface M ⊂R3 is minimal if and only if it can be locally expressed as the graph of a solution of •(1+ u x 2) u yy - 2 u x u y u xy + (1+ u y 2) u xx = 0 •Originally found in 1762 by Lagrange •In 1776, Jean Baptiste Meusnier discovered that it … We give a counterexample in R 2. The "first golden age" of minimal surfaces began. By Calabi’s correspondence, this also gives a family of explicit self-similar solutions for the minimal surface equation. Weierstrass and Enneper developed more useful representation formulas, firmly linking minimal surfaces to complex analysis and harmonic functions. By contrast, a spherical soap bubble encloses a region which has a different pressure from the exterior region, and as such does not have zero mean curvature. One might think that if the minimal surface equation had a solution on a smooth domain D ⊂ R n with boundary values φ, it would have a solution with boundary values tφ for all 0 ≤ t ≤ 1. ) if and only if f satisfies the minimal surface equation in divergence form: div grad(f) p 1 + jgrad(f)j2! = 0 Inthiscasewealsosaythat isaminimalsurface. My question is the following: since a geodesic is just a special case of a minimal surface, is there some analogous equation for the deviation vector field between two "infinitesimally nearby" minimal (or more generally, extremal) surfaces? o T do this, e w consider the set U g all tly (su cien smo oth) functions de ned on that are equal to g @. Minimal surface theory originates with Lagrange who in 1762 considered the variational problem of finding the surface z = z(x, y) of least area stretched across a given closed contour. Seiberg–Witten invariants and surface singularities Némethi, András and Nicolaescu, Liviu I, Geometry & Topology, 2002; What is a surface? Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. Ulrich Dierkes, Stefan Hildebrandt, and Friedrich Sauvigny. Exercise: (i) Verify the above derivation of the minimal surface equation. The above equation is called the minimal surface equation. etY another equivalent statement is that the surface is Minimal if and only if it's principal curvatures are equal in … 2 the surface M is generated by revolving about the x axis the curve segment y = f(x) joining P 1 - P 2. We provide a new and simpler derivation of this estimate and partly develop in the process some new techniques applicable to the study of hypersurfaces in general. uis minimal. Mémoire sur la courbure des surfaces. By the Young–Laplace equation, the mean curvature of a soap film is proportional to the difference in pressure between the sides. Then the Jacobi equation says that. If the projected Gauss map obeys the Cauchy–Riemann equations then either the trace vanishes or every point of M is umbilic, in which case it is a piece of a sphere. 8.3 Examples 140. One way to think of this "minimal energy; is that to imagine the surface as an elastic rubber membrane: the minimal shape is the one that in which the rubber membrane is the most relaxed. [citation needed] The endoplasmic reticulum, an important structure in cell biology, is proposed to be under evolutionary pressure to conform to a nontrivial minimal surface.[6]. Additionally, this makes minimal surfaces into the static solutions of mean curvature flow. Progress had been fairly slow until the middle of the century when the Björling problem was solved using complex methods. In 1776 Jean Baptiste Marie Meusnier discovered that the helicoid and catenoid satisfy the equation and that the differential expression corresponds to twice the mean curvature of the surface, concluding that surfaces with zero mean curvature are area-minimizing. We provide a new and simpler derivation of this estimate and partly develop in the process some new techniques applicable to the study of hypersurfaces in general. Savans, 10:477–510, 1785. Mém. This page was last edited on 27 February 2021, at 12:15. A surface in three dimensional space generated by revolving a plane curve about an axis in its plane. A direct implication of this definition and the maximum principle for harmonic functions is that there are no compact complete minimal surfaces in R3. The thin membrane that spans the wire boundary is a minimal surface; of all possible surfaces that span the boundary, it is the one with minimal energy. Miscellanea Taurinensia 2, 325(1):173{199, 1760. A direct implication of this definition is that every point on the surface is a saddle point with equal and opposite principal curvatures. In the art world, minimal surfaces have been extensively explored in the sculpture of Robert Engman (1927– ), Robert Longhurst (1949– ), and Charles O. Perry (1929–2011), among others. [7] In contrast to the event horizon, they represent a curvature-based approach to understanding black hole boundaries. Catalan proved in 1842/43 that the helicoid is the only ruled minimal surface. Example 3.3 Let be the graph of , a smooth function on . The partial differential equation in this definition was originally found in 1762 by Lagrange,[2] and Jean Baptiste Meusnier discovered in 1776 that it implied a vanishing mean curvature.[3]. Appendix A: Formulas from Multivariate Calculus 161. Soap films are minimal surfaces. If u is twice differentiable then integration by parts yields (2.2) or, equivalently, (2.3) div (a(\i'u)) = 0 This partial differential equation is known as the minimal surface equation. 92. a renewed interest in the theory of minimal surfaces [7]. Minimal surface theory originates with Lagrange who in 1762 considered the variational problem of finding the surface z = z(x, y) of least area stretched across a given closed contour. [4] Such discretizations are often used to approximate minimal surfaces numerically, even if no closed form expressions are known. So we get the minimal surface equation (MSE): div(ru p 1 + jruj2) We call the solution to this equation is minimal surface. Gaspard Monge and Legendre in 1795 derived representation formulas for the solution surfaces. An interior gradient bound for classical solutions of the minimal surface equation in n variables was established by Bombieri, De Giorgi, and Miranda in 1968. the positive mass conjecture, the Penrose conjecture) and three-manifold geometry (e.g. 303 0 obj <>/Filter/FlateDecode/ID[<9905AF4C536B704FAAAE36E66E929823>]/Index[189 129]/Info 188 0 R/Length 287/Prev 1231586/Root 190 0 R/Size 318/Type/XRef/W[1 2 1]>>stream Presented in 1776. Initiated by … 8.2 Derivation of MembraneWave Equation 138. 1 in the entire domain, the minimal surface problem is commonly known as Plateau’s Problem [4]. This is equivalent to having zero mean curvature (see definitions below). %PDF-1.5 %���� 2 f 11f 2! The minimal surface equation is the Euler equation for Plateau's problem in restricted, or nonparametric, form, which can be stated as follows [3, §18.9]: Let fix, y), a single-valued function defined on the boundary C of a simply connected region R in the x — y plane, represent the … ]�[�2UU���%,CR�-qT�4 mY.-����m���Cn�������u���;һm���C�j��+,W��e��{�aO�\C�t�R�Y^�I��\��Fw�+|N�Eaa��|/�����/�6=�` �6� Another cause was the verification by H. Karcher that the triply periodic minimal surfaces originally described empirically by Alan Schoen in 1970 actually exist. Phys. A simpler version of the equation is obtained by lineariza-tion: we assume that |Du|2 ˝ 1 and neglect it in the denominator. … In discrete differential geometry discrete minimal surfaces are studied: simplicial complexes of triangles that minimize their area under small perturbations of their vertex positions. If the soap film does not enclose a region, then this will make its mean curvature zero. Jung and Torquato [20] studied Stokes slow through triply porous media, whose interfaces are the triply periodic minimal surfaces, and explored whether the minimal surfaces are optimal for flow characteristics. This property establishes a connection with soap films; a soap film deformed to have a wire frame as boundary will minimize area. Acad. This property is local: there might exist regions in a minimal surface, together with other surfaces of smaller area which have the same boundary. Minimal surfaces are part of the generative design toolbox used by modern designers. 9.1 A Difficult Nonlinear Problem 149. Show that the Euler{Lagrange equation for the ‘surface area’ functional A[v] = Z p 1 + jrvj2 dx (v : !R) is the minimal surface equation div ru p 1 + jruj2 = 0: Problem 3. For a given constraint there may also exist several minimal surfaces with different areas (for example, see minimal surface of revolution): the standard definitions only relate to a local optimum, not a global optimum. the Smith conjecture, the Poincaré conjecture, the Thurston Geometrization Conjecture). [5], Minimal surfaces have become an area of intense scientific study, especially in the areas of molecular engineering and materials science, due to their anticipated applications in self-assembly of complex materials. Oxford University Press, Oxford, 2009. xxvi+785 pp. Using Monge's notations: p := ∂ f ∂ x; q := ∂ f ∂ y; r := ∂ 2 f ∂ x 2; s := ∂ 2 f ∂ x ∂ y; t := ∂ 2 f ∂ y 2; Where f ∈ C 2 ( Δ ⊂ R 2, R) is the minimal surface (any other function with the same values on the border of Δ has a bigger surface over it). )%-#+'����������������o`hdlbjfnaiemckg�����������������8�xeQa����͙=k��ӦN�. (1 + jr j 2) 1 = = 0: (2) This quasi-linear … 9 The KPIWave Equation 149. This definition uses that the mean curvature is half of the trace of the shape operator, which is linked to the derivatives of the Gauss map. with the classical derivation of the minimal surface equation as the Euler-Lagrange equation for the area functional, which is a certain PDE condition due to Lagrange circa 1762 de-scribing precisely which functions can have graphs which are minimal surfaces. u a ∇ a ( u b ∇ b η c) + R a b d a b d c u a u d η b = 0, where R a b c d is the Riemann tensor of the ambient space. ¼ >A7Y>hz á â ã ä Ï B6>AG6\8XY>/W XY:6>)i87958B`AG X \d^ XY:6>m^bZ6G6cAXn 0. The local least area and variational definitions allow extending minimal surfaces to other Riemannian manifolds than R3. derive the minimal surface equation by way of motivation. Lecture 7 Minimal Surface equations non-solvability strongly convex functional further regularity Consider minimal surface equation div √Du 1+|Du|2 = 0 in Ω u = ϕ on ∂Ω. The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Show that the Euler{Lagrange equation for E[v] = Z 1 2 jrvj 2 vf dx (v : !R) is Poisson’s equation u = f: Problem 2. "The classical theory of minimal surfaces", "Computing Discrete Minimal Surfaces and Their Conjugates", "Stacked endoplasmic reticulum sheets are connected by helicoidal membrane motifs", "Touching Soap Films - An introduction to minimal surfaces", 3D-XplorMath-J Homepage — Java program and applets for interactive mathematical visualisation, WebGL-based Gallery of rotatable/zoomable minimal surfaces, https://en.wikipedia.org/w/index.php?title=Minimal_surface&oldid=1009225491, Articles with unsourced statements from March 2019, Creative Commons Attribution-ShareAlike License. 1 = 0 from the minimal surface equation Lf= 1 + f2 2 f 11 2f 1f 2f 12 + 1 + f2 1 f 22 = 0: Bernstein™s way of computation is take derivative of the equation with respect to x 1 and eliminate the f 22 term in the resulting equation by the equation: 1 + f2 2 f 111 2f 1f 2f 121+ 1 + f2 1 f 221+2f 2f 21f 11! 8.1 Derivation of Minimal Surface Equation 137. Hence the catenoid is a minimal surface. %%EOF In the previous step, I have proven that for all h ∈ C 2: ∫ ∫ Δ p ∂ h ∂ x + q ∂ h ∂ y 1 + p 2 + q 2 d x d y = 0. Structures with minimal surfaces can be used as tents. While these were successfully used by Heinrich Scherk in 1830 to derive his surfaces, they were generally regarded as practically unusable. A famous example is the Olympiapark in Münich by Frei Otto, inspired by soap surfaces. The surface of revolution of least area. He derived the Euler–Lagrange equation for the solution Over surface meshes, a sixth-order geometric evolution equation was performed to obtain the minimal surface . This definition ties minimal surfaces to harmonic functions and potential theory. In the fields of general relativity and Lorentzian geometry, certain extensions and modifications of the notion of minimal surface, known as apparent horizons, are significant. 0 BIFURCATION FOR MINIMAL SURFACE EQUATION IN HYPERBOLIC 3-MANIFOLDS ZHENG HUANG, MARCELLO LUCIA, AND GABRIELLA TARANTELLO Abstract. 2. Essai d'une nouvelle methode pour determiner les maxima et les minima des formules integrales indefinies. Thus, we are led to Laplace’s equation divDu= 0. Jn J1 + IY'ul2. B. Meusnier. Generalisations and links to other fields. The minimal surface equation is nonlinear, and unfortunately rather hard to analyze. 1.1 Derivation of the Minimal Surface Equation Suppose that ˆRn is a bounded domain (that is, is open and connected). 9.2 Numerical Results 155. This has led to a rich menagerie of surface families and methods of deriving new surfaces from old, for example by adding handles or distorting them. The solution is a critical point or the minimizer of inf u| ∂Ω=ϕ Z Ω q 1+|Du|2. Example 3.4 The catenoid. 317 0 obj <>stream endstream endobj startxref Other important contributions came from Beltrami, Bonnet, Darboux, Lie, Riemann, Serret and Weingarten. Show that the Euler{Lagrange equation for the functional L W[v] = 1 2 Z R Z Rd jv 8 Minimal Surface and MembraneWave Equations 137. The fact that they are equivalent serves to demonstrate how minimal surface theory lies at the crossroads of several mathematical disciplines, especially differential geometry, calculus of variations, potential theory, complex analysis and mathematical physics.[1]. Expanding the minimal surface equation, and multiplying through by the factor (1 + jgrad(f)j2)3=2 weobtaintheequation (1 + f2 y)f xx+ (1 + f 2 x)f yy 2f xf yf xy= 0 In Fig. the sum of the principal curvatures at each point is zero. Triply Periodic Minimal Surfaces A minimal surface is a surface that is locally area-minimizing, that is, a small piece has the smallest possible area for a surface spanning the boundary of that piece. Question. Minimal surfaces necessarily have zero mean curvature, i.e. Oxford Mathematical Monographs. 189 0 obj <> endobj Fix ˚: @!R, and introduce L(;˚) := fu2C0;1(); uj @ = ˚g; (1.1) the set of Lipschitz functions on whose restriction to @ is ˚. Bernstein's problem and Robert Osserman's work on complete minimal surfaces of finite total curvature were also important. J. L. Lagrange. We prove several results in these directions. He did not succeed in finding any solution beyond the plane. He derived the Euler–Lagrange equation for the solution. Brownian motion on a minimal surface leads to probabilistic proofs of several theorems on minimal surfaces. Between 1925 and 1950 minimal surface theory revived, now mainly aimed at nonparametric minimal surfaces. This not only stimulated new work on using the old parametric methods, but also demonstrated the importance of computer graphics to visualise the studied surfaces and numerical methods to solve the "period problem" (when using the conjugate surface method to determine surface patches that can be assembled into a larger symmetric surface, certain parameters need to be numerically matched to produce an embedded surface).

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