Minimal Surface

Minimal surfaces are defined as surfaces with zero Mean Curvature, and therefore satisfy Lagrange's Equation

Euler proved that a minimal surface is planar Iff its Gaussian Curvature is zero at every point sothat it is locally Saddle-shaped. The Existence of a solution to the general case was independently proven byDouglas (1931) and Radó (1933), although their analysis could not exclude the possibility of singularities. Osserman (1970)and Gulliver (1973) showed that a minimizing solution cannot have singularities.

The only known complete (boundaryless), embedded (no self-intersections) minimal surfaces of finite topology known for 200years were the Catenoid, Helicoid, and Plane. Hoffman discovered a three-ended Genus 1minimal embedded surface, and demonstrated the existence of an infinite number of such surfaces. A four-ended embeddedminimal surface has also been found. L. Bers proved that any finite isolated Singularity of a single-valuedparameterized minimal surface is removable.

A surface can be parameterized using a Isothermal Parameterization. Such a parameterization is minimal if thecoordinate functions are Harmonic, i.e., are Analytic. A minimal surface can therefore be defined by a triple of Analytic Functions suchthat . The Real parameterization is then obtained as

(1)

			(2)
			(3)
			(4)

(5)

Minimal Surfaces

Dickson, S. ``Minimal Surfaces.'' Mathematica J. 1, 38-40, 1990.

Dierkes, U.; Hildebrandt, S.; Küster, A.; and Wohlraub, O. Minimal Surfaces, 2 vols. Vol. 1: Boundary Value Problems. Vol. 2: Boundary Regularity. Springer-Verlag, 1992.

do Carmo, M. P. ``Minimal Surfaces.'' §3.5 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 41-43, 1986.

Douglas, J. ``Solution of the Problem of Plateau.'' Trans. Amer. Math. Soc. 33, 263-321, 1931.

Fischer, G. (Ed.). Plates 93 and 96 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 89 and 96, 1986.

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, p. 280, 1993.

Gulliver, R. ``Regularity of Minimizing Surfaces of Prescribed Mean Curvature.'' Ann. Math. 97, 275-305, 1973.

Hoffman, D. ``The Computer-Aided Discovery of New Embedded Minimal Surfaces.'' Math. Intell. 9, 8-21, 1987.

Hoffman, D. and Meeks, W. H. III. The Global Theory of Properly Embedded Minimal Surfaces. Amherst, MA: University of Massachusetts, 1987.

Lagrange. ``Essai d'une nouvelle méthode pour déterminer les maxima et les minima des formules intégrales indéfinies.'' 1776.

Meusnier, J. B. ``Mémoire sur la courbure des surfaces.'' Mém. des savans étrangers 10 (lu 1776), 477-510, 1785.

Nitsche, J. C. C. Introduction to Minimal Surfaces. Cambridge, England: Cambridge University Press, 1989.

Osserman, R. A Survey of Minimal Surfaces. New York: Van Nostrand Reinhold, 1969.

Osserman, R. ``A Proof of the Regularity Everywhere of the Classical Solution to Plateau's Problem.'' Ann. Math. 91, 550-569, 1970.

Radó, T. ``On the Problem of Plateau.'' Ergeben. d. Math. u. ihrer Grenzgebiete. Berlin: Springer-Verlag, 1933.

• Parallelizable
• Parallelogram
• Parallelogram Illusion
• Parallelogram Law
• Parallelohedron
• Parallelotope
• Parallel Postulate
• Parallel (Surface of Revolution)
• Paralogic Triangles
• Parameter
• Parameterization
• Parameter (Quadric)
• Parametric Latitude
• Parametric Test
• Pareto Distribution
• Parity
• Parity Constant
• Parking Constant
• Parodi's Theorem
• Parry Circle
• Parry Point
• Parseval's Integral
• Parseval's Relation
• Parseval's Theorem
• Partial Derivative