![]() ![]() ![]() the positive mass conjecture, the Penrose conjecture) and three-manifold geometry (e.g. 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.Ĭurrently the theory of minimal surfaces has diversified to minimal submanifolds in other ambient geometries, becoming relevant to mathematical physics (e.g. Karcher that the triply periodic minimal surfaces originally described empirically by Alan Schoen in 1970 actually exist. 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). 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 R 3 of finite topological type. Bernstein's problem and Robert Osserman's work on complete minimal surfaces of finite total curvature were also important.Īnother revival began in the 1980s. The complete solution of the Plateau problem by Jesse Douglas and Tibor Radó was a major milestone. Other important contributions came from Beltrami, Bonnet, Darboux, Lie, Riemann, Serret and Weingarten.īetween 19 minimal surface theory revived, now mainly aimed at nonparametric minimal surfaces. Weierstrass and Enneper developed more useful representation formulas, firmly linking minimal surfaces to complex analysis and harmonic functions. Schwarz found the solution of the Plateau problem for a regular quadrilateral in 1865 and for a general quadrilateral in 1867 (allowing the construction of his periodic surface families) using complex methods. The "first golden age" of minimal surfaces began. Progress had been fairly slow until the middle of the century when the Björling problem was solved using complex methods. Catalan proved in 1842/43 that the helicoid is the only ruled minimal surface. While these were successfully used by Heinrich Scherk in 1830 to derive his surfaces, they were generally regarded as practically unusable. Gaspard Monge and Legendre in 1795 derived representation formulas for the solution surfaces. 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.ĭifferential equation definition: A surface M ⊂ R 3 is minimal if and only if it can be locally expressed as the graph of a solution of ( 1 + u x 2 ) u y y − 2 u x u y u x y + ( 1 + u y 2 ) u x x = 0 If the soap film does not enclose a region, then this will make its mean curvature zero. By the Young–Laplace equation, the mean curvature of a soap film is proportional to the difference in pressure between the sides. Additionally, this makes minimal surfaces into the static solutions of mean curvature flow. Mean curvature definition: A surface M ⊂ R 3 is minimal if and only if its mean curvature is equal to zero at all points.Ī direct implication of this definition is that every point on the surface is a saddle point with equal and opposite principal curvatures. On a minimal surface, the curvature along the principal curvature planes are equal and opposite at every point. This definition makes minimal surfaces a 2-dimensional analogue to geodesics, which are analogously defined as critical points of the length functional. Variational definition: A surface M ⊂ R 3 is minimal if and only if it is a critical point of the area functional for all compactly supported variations. This property establishes a connection with soap films a soap film deformed to have a wire frame as boundary will minimize area. This property is local: there might exist regions in a minimal surface, together with other surfaces of smaller area which have the same boundary. Local least area definition: A surface M ⊂ R 3 is minimal if and only if every point p ∈ M has a neighbourhood, bounded by a simple closed curve, which has the least area among all surfaces having the same boundary. 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. Minimal surfaces can be defined in several equivalent ways in R 3. ![]() ![]() While any small change of the surface increases its area, there exist other surfaces with the same boundary with a smaller total area. ![]()
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