Minimal Surfaces

Abstract

Since the last century, the name minimal surfaces has been applied to surfaces of vanishing mean curvature, because the condition

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGib % Gaeyypa0JaaGimaaaa!3A36!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$H = 0$$

will necessarily be satisfied by surfaces which minimize area within a given boundary configuration. This was implicitly proved by Lagrange for nonparametric surfaces in 1760, and then by Meusnier in 1776 who used the analytic expression for the mean curvature and determined two minimal surfaces, the catenoid and the helicoid. (The notion of mean curvature was introduced by Young [1] and Laplace [1], but usually it is ascribed to Sophie Germain [1].) In Section 2.1 we shall derive an expression for the first variation of area with respect to general variations of a given surface. From this expression we obtain the equation H = 0 as necessary condition for stationary surfaces of the area functional, and we also demonstrate that solutions of the free boundary problem meet their supporting surfaces at a right angle.