By Robert Osserman

ISBN-10: 0486495140

ISBN-13: 9780486495149

Divided into 12 sections, this article explores parametric and nonparametric surfaces, surfaces that reduce region, isothermal parameters on surfaces, Bernstein's theorem and masses extra. Revised version comprises fabric on minimum surfaces in relativity and topology, and up to date paintings on Plateau's challenge and on isoperimetric inequalities. 1969 variation.

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**Extra info for A Survey of Minimal Surfaces (Dover Phoenix Editions)**

**Example text**

The same is true if one sets f3-if4 = g(z) . 14) Now by Theorem S. 15) (k4 = -d, d= 1 + c2 We consider two cases. 16) a=0, b=+1. 16) implies 94 = ± i93 which is equivalent to f3 + if4 an analytic function of z or z . 14). 17) (q 3 + d, Thus, each of the factors on the left is different from zero. In particular, the function 3- ic4 is an entire function which never vanishes, and therefore is of the form 03-rc4 = eH(w) for some entire function 11(w). 18) 03 = 21- (e"'- de Hf'"1) , 04 = 2 (eH(w) + de-H(w)) We can thus describe explicitly in the case n = 4 all solutions of the minimal surface equation which are valid in the whole x1, x2plane.

We then have an associated simply-connected minimal surface S, called the universal covering surface of S, defined by the composed map x(rr(p)) : M -, E". It follows that S is regular, if and only if S is regular, and S is complete if and only if S is complete. Thus, many questions concerning minimal surfaces may be settled by considering only simply-connected minimal surfaces. In that case we have the following important simplification. 3. Every simply-connected minimal surface S has a reparametrization in the form x(C) : D - E", where D is either the unit disk, CI < 1, or the entire c-plane.

However, if V2 is small enough so that V2 C A, then x(u) = x(u) on r, so that Xx will be a surface with the same boundary as E. The assumption that E minimizes area implies that A(A) > A(0) for all a, whence A'(0) = 0. 6), and the assertion is proved. Thus minimal surfaces arose originally in connection with minimizing area, and it is from this connection that they derived their name. However, as we shall see, they also arise naturally in a number of other connections, and many of their most important properties are totally unrelated to questions of area.

### A Survey of Minimal Surfaces (Dover Phoenix Editions) by Robert Osserman

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