The Thread Problem
The problem studied in this chapter is another generalization of the isoperimetric problem related to minimal surfaces. Consider a fixed arc Γ with endpoints P 1 and P 2 connected by a movable arc Σ of fixed length. One may conceive Γ as a thin rigid wire, at the ends of which a thin inextensible thread Σ is fastened. Then the thread problem is to determine a minimal surface minimizing area among all surfaces bounded by the boundary configuration 〈Γ,Σ〉. The particular feature distinguishing this problem from the ordinary Plateau problem is the movability of the arc Σ.
First, several variants of the thread problem are described, and some experimental solutions are depicted. Most of these questions have not yet been treated mathematically, that is, no existence proof can be found in the literature. Here the mathematical formulation of the thread problem in the simplest case is stated, and then the existence proof given by H.W. Alt for this case is outlined. The main difficulty to be overcome is that one can no longer preassign the topological type of the parameter domain on which the desired minimizer will be defined. Finally the regularity of the movable part Σ of the boundary of the area-minimizing surface will be investigated. The main result is that Σ is a regular real analytic arc of constant curvature.
KeywordsMinimal Surface Branch Point Constant Curvature Parameter Domain Existence Proof
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