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 variation comprises 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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Extra resources for A survey of minimal surfaces
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. But an LlX· N arbitrary normal vector to S, we have a2 x au f . N + a2 x . 4). 5). 5) is of interest in other connections and in particular in the study of surfaces of constant mean curvature.
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. be the region bounded by ï. I f the surface xk f(x1, x2) over ô.
A, and N(u) C 1 may assume N "' N(a) H(N) fo such that O. We By Lemma 2 . 2 we can find a neighborhood V1 of in V 1, such that N(u) is normal to S at x(u). 6) will be strictly positi ve. , then i(u) l = = �À w ill be a surface with the same bound ary as�. The assump tion that � minimizes area implies that A(À) 2: A(O) for all À, whence A '(0) O. 6), and the assertion is proved. Thus minimal surfaces arose originally in connection with mini mizing area, and it is from this connection that they derived their name.
A survey of minimal surfaces by Robert Osserman