By Robert Osserman
Divided into 12 sections, this article explores parametric and nonparametric surfaces, surfaces that reduce zone, isothermal parameters on surfaces, Bernstein's theorem and masses extra. Revised variation 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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Extra resources for A survey of minimal surfaces
Where M is a positive constant. 4) for and recalling the formula (1. ijdu1 du2 � for the rate of change of area as a function of À. 7) k With this notation, if we choose our family of surfaces ting h ( u) = l, th en formula (3. 6) reduces to A '(0) -2 • SÀ by set 1! H(N)dA k which provides an interesting interpretation of the quantity H(N). We now return to our original problem, and we make the follow in order for I. to minimize area, its mean curvature must be identically zero. This follows immed iately from (3.
1) · ax a ui _ 0, Î 1, 2. Differentiating this equation yields (3. 2) [ We now consider an arbitrary function h (u) •ach real number À we form the surface SÀ : e find R (u) == x (u) + l Àh (u) N(u), c2 in D, and for u l D. SURFACES THAT MJNJMIZE AREA ax. 2) g . l] . = ax . ax aÛ. au. l 1 = g l] . - 2Ahb l] . (N) + 2 A c IJ . where c l].. is a continuous function of u in D. 3) where and a2 is a continuous function of ul' u2, À for u in D. , the 0 < €. and u € �. In other words, for for det �ii> lAI lAI surfaces I,À defined by restricting x(u) to � are all regular sur faces.
N, we may represent the -> = . surface in terms of the parameters Ç 1 , Ç2 • We find 2 k = 3, ax2 a ç-2 - , n; . . 2 W + 1+ IPI JW 3, . . , n. It follows that with respect to the parameters Ç l' Ç 2 , we have w2 . 2 , 2W+2+IPI +Iql so that Çl' Ç2 are isothermal coordinates . COROLLARY. Let xk 2 + fk(xl' x 2 ), k = 3, , n, define a mini mal surface in non-parametric form. Then the fk are real analytic functions of x1, x2 • . 33 ISOTHERMAL PARAMETERS Proof: In a nei ghborhood of each point we can introduce the m ap (4.