By Y. Matsushita

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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.

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.

10) I 10 0 . 3. Let x(u) define a regular minimal surface, with ul, u2 isothermal parameters. 10). 10) in a simply-connected domain D. 6) are valid. 2. 2. 10). 11), these functions determine the xk up to additive constants, and the surface is therefore determined up to a translation. 31 ISOTHERMAL PARAMETERS The preceding results are all based on the assumption that the surface can be represented locally in terms of isothermal parameters. However, the existence of such parameters is not at all obvious, and in the case of C1-surfaces it is not even always true.