So we get the minimal surface equation (MSE): div(ru p 1 + jruj2) We call the solution to this equation is minimal surface. We prove several results in these directions. B. Meusnier. A surface in three dimensional space generated by revolving a plane curve about an axis in its plane. 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 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. 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. By Calabi’s correspondence, this also gives a family of explicit self-similar solutions for the minimal surface equation. 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. ¼ >A7Y>hz á â ã ä Ï B6>AG6\8XY>/W XY:6>)i87958B`AG X \d^ XY:6>m^bZ6G6cAXnstream An equivalent statement is that a surface SˆR3is Minimal if and only if every point p2Shas a neighbourhood with least-area relative to its boundary. 2 f 11f 2! The minimal surface equation is nonlinear, and unfortunately rather hard to analyze. A direct implication of this definition is that every point on the surface is a saddle point with equal and opposite principal curvatures. 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. Derivation of the Partial Differential Equation Given a parametric surface X(u,v) = hx(u,v),y(u,v),z(u,v)i with parameter domain D, ... For a minimal surface, the eigenvalues of the matrix S are opposites of one another, and thus Structures with minimal surfaces can be used as tents. The criterion for the existence of a minimal surface in $ E ^ {3} $ with a given metric is given in the following theorem of Ricci: For a given metric $ ds ^ {2} $ to be isometric to the metric of some minimal surface in $ E ^ {3} $ it is necessary and sufficient that its curvature $ K $ be non-positive and that at the points where $ K < 0 $ the metric $ d \sigma ^ {2} = \sqrt {- K } ds ^ {2} $ be Euclidean. endstream endobj startxref In architecture there has been much interest in tensile structures, which are closely related to minimal surfaces. 8.3 Examples 140. 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. Mém. Catalan proved in 1842/43 that the helicoid is the only ruled minimal surface. Another revival began in the 1980s. Presented in 1776. 92. 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. In this paper, we consider the existence of self-similar solution for a class of zero mean curvature equations including the Born–Infeld equation, the membrane equation and maximal surface equation. Phys. Show that the Euler{Lagrange equation for the functional L W[v] = 1 2 Z R Z Rd jv The local least area and variational definitions allow extending minimal surfaces to other Riemannian manifolds than R3. 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? 9.1 A Difficult Nonlinear Problem 149. 189 0 obj <> endobj uis minimal. This property is local: there might exist regions in a minimal surface, together with other surfaces of smaller area which have the same boundary. 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. Seiberg–Witten invariants and surface singularities Némethi, András and Nicolaescu, Liviu I, Geometry & Topology, 2002; What is a surface? Other important contributions came from Beltrami, Bonnet, Darboux, Lie, Riemann, Serret and Weingarten. Progress had been fairly slow until the middle of the century when the Björling problem was solved using complex methods. Definition 3.2 A smooth surface with vanishing mean curvature is called a minimal surface. Classical examples of minimal surfaces include: Surfaces from the 19th century golden age include: Minimal surfaces can be defined in other manifolds than R3, such as hyperbolic space, higher-dimensional spaces or Riemannian manifolds. 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. par div. a renewed interest in the theory of minimal surfaces [7]. Yvonne Choquet-Bruhat. 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). Question. Another cause was the verification by H. Karcher that the triply periodic minimal surfaces originally described empirically by Alan Schoen in 1970 actually exist. . 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. 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. 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 … By the Young–Laplace equation, the mean curvature of a soap film is proportional to the difference in pressure between the sides. Minimal surfaces necessarily have zero mean curvature, i.e. 2. 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 This property establishes a connection with soap films; a soap film deformed to have a wire frame as boundary will minimize area. 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. Example 3.3 Let be the graph of , a smooth function on . One cause was the discovery in 1982 by Celso Costa of a surface that disproved the conjecture that the plane, the catenoid, and the helicoid are the only complete embedded minimal surfaces in R3 of finite topological type. the positive mass conjecture, the Penrose conjecture) and three-manifold geometry (e.g. %%EOF 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. = 0 Inthiscasewealsosaythat isaminimalsurface. 2. Between 1925 and 1950 minimal surface theory revived, now mainly aimed at nonparametric minimal surfaces. The complete solution of the Plateau problem by Jesse Douglas and Tibor Radó was a major milestone. The definition of minimal surfaces can be generalized/extended to cover constant-mean-curvature surfaces: surfaces with a constant mean curvature, which need not equal zero. [7] In contrast to the event horizon, they represent a curvature-based approach to understanding black hole boundaries. We give a counterexample in R 2. A minimal surface is a surface each point of which has a neighborhood that is a surface of minimal area among the surfaces with the same boundary as the boundary of the neighborhood. 0 The above equation is called the minimal surface equation. Bernstein's problem and Robert Osserman's work on complete minimal surfaces of finite total curvature were also important. In mathematics, a minimal surface is a surface that locally minimizes its area. h�b```"Kv�" ���,�260�X�}_�xևG���J�s�U��a�����������@�������������/ ($,"*&.! But the integrand F (p) = q 1+|p|2 is not strongly convex, that is D2F δI, only D2F > 0. 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]. 8.4 Problems 142. 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. The loss of strong convexityor convexity causes non-solvability, or non 2 the surface M is generated by revolving about the x axis the curve segment y = f(x) joining P 1 - P 2. Weierstrass and Enneper developed more useful representation formulas, firmly linking minimal surfaces to complex analysis and harmonic functions. 9 The KPIWave Equation 149. )%-#+'����������������o`hdlbjfnaiemckg�����������������8�xeQa����͙=k��ӦN�. 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 Paris, prés. While these were successfully used by Heinrich Scherk in 1830 to derive his surfaces, they were generally regarded as practically unusable. He derived the Euler–Lagrange equation for the solution. Gaspard Monge and Legendre in 1795 derived representation formulas for the solution surfaces. Thus, we are led to Laplace’s equation divDu= 0. 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. 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. … Savans, 10:477–510, 1785. Mémoire sur la courbure des surfaces. In discrete differential geometry discrete minimal surfaces are studied: simplicial complexes of triangles that minimize their area under small perturbations of their vertex positions. Acad. 1.1 Derivation of the Minimal Surface Equation Suppose that ˆRn is a bounded domain (that is, is open and connected). 8.1 Derivation of Minimal Surface Equation 137. In Fig. This is equivalent to having zero mean curvature (see definitions below). A classical result from the calculus of ariations v asserts that if u is a minimiser of A (u) in U g, then it satis es the Euler{Lagrange equation r u. BIFURCATION FOR MINIMAL SURFACE EQUATION IN HYPERBOLIC 3-MANIFOLDS ZHENG HUANG, MARCELLO LUCIA, AND GABRIELLA TARANTELLO Abstract. 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. + f 1f 21 f 12+2f 1f 11f 22 = 0 and 1 + f2 2 f 111 2f 1f 11f 11 1 + f2 1 2f 1f 2 f 121 2f 1f 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].