By Robert Osserman

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 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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Tri ples - a local surface in i sotherm al p arameters. By Lem m a 4 . 4 the union Oa equals M, so that A i s an atlas for M, and by Lemma 4 . 5 e ach Fa - 1 o F{3 i s conformai wherever defined, so th at À defines a conformai structure on M. t - o f the - 46 A SUR VEY O F MINIMAL SURFACES Let us note that the introd uction of a conformai structure in thi s way m ay be carried out in gre at generality, since low-order d ifferentiability conditions on S guarantee the existence of local i sothermal p arameters; however, the proof of their exi stence is far more difficult in the general case.

Mum, hence it would be constant , cont radicting the assumption hat the map x(p) is non-constant. + Finally, concerning the study of generali zed minima l sur faces , let us note that precisely properties of the branch points 1themselves may be an object of investigation. See, for exemple, �ers [ 2] and Chen [ 1]. For the sake o f brevity we make the fol lowing convention. We $hall suppress the adj ecti ves "generalized" and "reguler, " and i•see Appendtx 3, Section 1. 49 P ARAMETRIC SURFACES: GLOBAL TH EORY we shall refer simply to "minimal surfaces" except in tho se cases where either the statement would not be true without suitably qual i fying it, or else where we wish to emphasize the fact that the sur face s in question are "re gular" or " generalized.

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 . 10). ) We shall see tha t Equations (3. 1 6) have equal y important consequences in the case o f arbitrary n.