Differential Geometry

Download A Survey of Minimal Surfaces (Dover Phoenix Editions) by Robert Osserman PDF

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 variation.

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6), k = 1, 2, 3, we see that 41 and 952 are constant since xl and x2 are linear functions of ul, u2. 9), 953 must also be constant. This means that x3 has constant gradient with respect to u1, u2, hence also with respect to x1, x2. Thus f(xi, x2) = Axi + Bx2 + C. Conversely, if f is of this form, it is easy to write down an explicit linear transformation yielding isothermal coordinates; for example, x1 = AAul + But, x2 = ABu 1- Au2, where X2 = 1/(1 + A2 + B2). 1. (Osserman [71). 8) in the whole x1, x2-plane.

A generalized minimal surface cannot be compact. Proof: Let S be a generalized minimal surface defined by a map x(p) : M E". Then each coordinate function xk(p) is harmonic on M, and if M were compact xk(p) would attain its maximum, hence it would be constant, contradicting the assumption that the map x(p) is non-constant. 1 Finally, concerning the study of generalized minimal surfaces, let us note that precisely properties of the branch points themselves may be an object of investigation. See, for example, Bers [2] and Chen [1].

2) iij = az au. I where , ax au. = g.. - 2Ahb1j (N) + A2c ij J is a continuous function of u in D. 4) a0 = det gij, al = -2h(g11b22(N) + g22b11(N)-2g12b12(N)) and a2 is a continuous function of ul, u2, A for u in D. As a first consequence of this formula, using the fact that S is regular we deduce that a0 has a positive minimum on A, and since al and a2 are continuous in D, there exists E > 0 such that det glj > 0 for JAJ < E and u c 0. In other words, for JAl < E, the surfaces Xx defined by restricting a`c(u) to A are all regular sur faces.

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