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 comprises fabric on minimum surfaces in relativity and topology, and up-to-date paintings on Plateau's challenge and on isoperimetric inequalities. 1969 version.

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**Example text**

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.

Many of the basic quantities considered in su rface theory s impl i f y considerabl y when referred to isot herm al parameters. 3) (4. 2) we have clet g 1] . 4) We also have the following useful formula for the Laplac ian of the coordinate vector of an arbitrary surface. 1. S be defined by (u) C2 where u l' u 2 are isothermal parameters. 5) where H is the mean curvature vector. 1) for isothermal parameters may be written in the form 0 . Differentiating the first of the se with respect to u 1, and the second with respect to u2 yields whence 0 Sim ilarly, different iating the first equation with respect to the second with respect to Thus Llx if N is u1 u2 and yields is a vector perpendicular to the tangent plane to S.

COROLL ARY 2. 8) in the whole plane must be constant (for arbitrary n) . COROL LARY 3. 8) in the } 2 whole x l ' x 2 -plane, and let S be the surface defined by k = 3, . 8) . 7). 9) k = 3, . . , n are analytic functions of u 1 satisfy n (S. 10) I k= 3 + iu 2 in the whole u l ' u 2 -plane and cf>; '-" - l - c 2, c = a- i b . Converse/y, gi ven any complex constant c = a - ib wi th b > 0, and given any entire functions (f> 3, . , if; of u 1 + iu2 satisfying n ( S . 1 0) , equations (S. 9) may be used to defi ne harmonie fun etions .