By Robert Osserman

Divided into 12 sections, this article explores parametric and nonparametric surfaces, surfaces that reduce quarter, isothermal parameters on surfaces, Bernstein's theorem and lots more and plenty extra. Revised version contains 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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3. {Reflection principle). = = . . = = in the full disk, by the r eflection principle for harmonie functions. T hus the functio n s k= l, . . ,n - 1 are analytic in the full disk and are pure i maginary on the real axis. MINIMAL SURFACES WITH BOUNDARY 55 By the equation 4>; we see that 4>; n- 1 = - I k= 1 4>; extends continuously to the real axis and has non- negative real values th ere. lt follows that 4> n extends continuous 1 xn a xn ;au2 0 ly to the real axis and has real values there. By i ntegration, extends continuously to the real axis, and satisfies there.

9). If we expand out the coefficient of p in the second term, we find the exp ression (3. _ w3 [(p q)q -( 1 + 1 ql 2 )p] · • [(l + 1 ql 2 )r - p q)s 1 2 )t 2( + ( + IPI ] which vanishes by (3. 10). Interchanging see that the coefficient of q • p and q, x 1 and x2 , we in the th ird term vanishes a lso, thus roving (3. 14). ln the process we have also shown that the two equations b . 16) are satisfied by every solution of the minimal surface equation • 3 . 10). These equat ions have long been known in the case n = 3, and the fact that they are in divergence form al lows one to derive many consequences which are not immediate from (3 .

3) applied to this function. Th us Lem m a S . 3 . LEMM A s . s . L et f{x r x 2) t c 1 in a domain D, where f is real-valued. N ecessary and sufficient that the surface S: x3 f{x 1 , x 2 ) lie on a plane is that there exist a nonsingular linear = transformation { u 1 , u 2 ) parameters on S. ""* {x 1 , x 2) such tha t u 1 , u 2 are isothermal Proof: Suppose such parameters u 1, u 2 exist. 6), k 1, 2, 3, we see that k constant sin ce x and x 2 are linear functions of 1 ( 4. 9), cp 3 must also be constant.