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Linearization and explicit solutions of the minimal surface equation

Publicacions Matemàtiques
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  • Mathematics


Publicacions Matemátiques, Vol 36 (1992), 39-46 . LINEARIZATION AND EXPLICIT SOLUTIONS OF THE MINIMAL SURFACE EQUATION A bstract ALEXANDER G. REZNIKOV We show that the apparatus of support functions, usually used in convex surfaces theory, leads to the linear equation Ah+2h = 0 de- scribing locally germs of minimal surfaces . Here A is the Laplace- Beltrami operator on the standard two-dimensional sphere . It explains the existente of the sum operation of minimal sur- faces, introduced recently. In 4-dimensional space the equation Oh + 2h = 0 becomes inequality wherever the Gauss curvature of a minimal hypersurface is nonzero . 0. Introduction Recently great progress was achieved in the investigation and con- struction of examples of minimal surfaces in R3 [1]-[3] . The Gauss map usually plays a significant role and its singularities in a sense control topology if the surface is complete [4] . It was also noticed [5], [6] that there exists a "sum" operation Ml + M2 for two minirnal surfaces Ml, M2 . It may seem to be strange, for the usual form of the minimal surface equation is essentially nonlinear . True, given a conformal minimal map R2 D U x > M C R3 we have a linear equation áx = 0 [7] . However, the condition of conformality is nonlinear itself. In this paper, we show that apparatus of support functions usually used in convex surface theory leads to the linear and completely inte- gradle equation of minimal surfaces in R3 . We are able to write down an explicit formula describing locally all minimal surfaces with nonvanishing curvature which is quite different from the Weierstrass description . We hope our method will be useful in global problems, too . It automatically implies the existente of the sum operation . The main part of this work was done during the author's visit to Lithuania in 1987 . I wish to thank Professor F . Weiksa for fruitful dis- cussions . I also wish to thank the referee for his very valuable remarks, in particular, for indicating to me that the re

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