By Ulrich Dierkes

ISBN-10: 3642117058

ISBN-13: 9783642117053

Many houses of minimum surfaces are of an international nature, and this can be already precise for the implications handled within the first volumes of the treatise. half I of the current booklet will be seen as an extension of those effects. for example, the 1st chapters take care of lifestyles, regularity and forte theorems for minimum surfaces with partly loose limitations. the following one of many major beneficial properties is the opportunity of "edge-crawling" alongside loose components of the boundary. The 3rd bankruptcy offers with a priori estimates for minimum surfaces in better dimensions and for minimizers of singular integrals relating to the world useful. particularly, a ways achieving Bernstein theorems are derived. the second one a part of the publication includes what one may perhaps justly name a "global thought of minimum surfaces" as estimated by way of Smale. First, the Douglas challenge is taken care of anew through the use of Teichmüller thought. Secondly, quite a few index theorems for minimum theorems are derived, and their results for the distance of options to Plateau´s challenge are mentioned. eventually, a topological method of minimum surfaces through Fredholm vector fields within the spirit of Smale is presented.

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**Sample text**

The vertices gj := σ(tj ) of Σ0 lie in the interior of Σ. Secondly we suppose that σ (s) and σ (s) tend to limits as s → ∞ and s → −∞ (whence σ (s) → 0 as s → ±∞). 6 in Vol. 2. If σ is only piecewise smooth we use the one-sided tangents σ (s + 0) and σ (s − 0) instead of σ (s) at s = t1 , . . , tl . We suppose that the tangents σ (tj + 0) and σ (tj − 0) enclose an angle βj = 0, π, 2π at any vertex qj = σ(tj ) of Σ, and that Γ meets Σ0 nontangentially at p1 and p2 . Let t : Σ0 → S 1 and ν : Σ0 → S 1 be two piecewise continuous ﬁelds of unit vectors along Σ0 such that t(σ(s)) = σ (s) and ν(σ(s)) · σ (s) = 0 provided that σ is diﬀerentiable at the value s.

Then only the following three cases can occur: (I) x0 = 0, and I2 consists of a single point u0 ; (II) x0 = 0, and I2 is a closed interval of positive length; (III) x0 > 0, that is, I2 is empty, and there is exactly one point u0 in I such that x0 = x(u0 ). Consequently, we have x(u) > x0 for u ∈ I with u = u0 . Remark. 2. If we introduce the new supporting surface Sε = {(x, y, z) ∈ R3 : y = 0, x ≥ −ε}, ε > 0, for some suﬃciently small ε > 0 as well as the new coordinates ξ = x + ε, η = y, ζ = z, a surface X(w) = (x(w), y(w), z(w)) of type I is transformed into a surface Ξ(w) = (ξ(w), η(w), ζ(w)) of type III.

Theorem 1 (Representation theorem). Let X be a stationary minimal surface in C(Γ, S), and let x0 be the lowest x-level of the free trace of X, that is, x0 := min{x(u) : u ∈ I}. Moreover, denote by D = D(x0 ) the twodimensional domain in the x, y-plane which is obtained from the interior of the orthogonal projection γ of Γ by slicing this interior along the x-axis from ˆ of the slit domain D, both x = x0 to x = a. In deﬁning the boundary ∂D borders of the slit x0 < x ≤ a will appear, with opposite orientation.

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