Abstract
Since the last century, the name minimal surfaces has been applied to surfaces of vanishing mean curvature, because the condition
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Dierkes, U., Hildebrandt, S., Küster, A., Wohlrab, O. (1992). Minimal Surfaces. In: Minimal Surfaces I. Grundlehren der mathematischen Wissenschaften, vol 295. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02791-2_3
Download citation
DOI: https://doi.org/10.1007/978-3-662-02791-2_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-02793-6
Online ISBN: 978-3-662-02791-2
eBook Packages: Springer Book Archive