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 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 version.

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15) E - d, d ± i, then equation W e consider two cases. F i rst, if c (5. 10) re d u ces to a= (5. 16) Thus the t r ansfo rm ation (5. ± z. Th us the case c = lutions (5. 14). (5. 13) and 0, b = + l. either the ident i ty transformation and equation (5. 16) i mplies

Then the functions xk(u), for k 1, . . , n- 1, m a y be extended by setting xk(u 1 , 0) 0, xk (u 1, u 2) = - xk (u 1, - u 2). 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.

12) One often u se s the notation W (3. 13) ydet g ii or non -parametric surfaces. Suppose now that we make a variat ion in our surface, setting k = 3, , . . l where n, À is a real number, and hk c c1 in the domain of defin ition 'v of the fk" In vector notation, setting h (h3, , hn) we have • • • Ê = f + Àh, p W2 where + 2ÀX + À2 Y , 25 SURFACES THAT MINIMIZE AREA and Y is cont inuons in xl' x2• It follows that where z is again continuo ns. We now consider a closed curve ï in the domain o f definition of f(x1, x2), and let ô.