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Computing minimal surfaces via level set curvature flow. (English) Zbl 0786.65015

A new method is presented for solving numerically Plateau’s problem of constructing a minimal surface with a prescribed boundary curve. The surface is represented as a level set of a global real function ${\Phi }$ in ${ℝ}^{3}$. The function ${\Phi }$ depends additionally on a parameter $t$ and evolves according to mean curvature flow until a steady state is reached. The surface is attached to the fixed boundary curve by interpolatory conditions. The basic algorithm as well as procedures to avoid collapse near the boundary and to make the algorithm fast, are described in detail.

The topology of the level set of ${\Phi }$ can change easily during the iteration. This offers the opportunity to determine the topological type of the minimal surface automatically. This is a big advantage over existing methods which are either incapable or require ad hoc intervention to change topology. A series of fourteen calculated minimal surfaces shown in grey scale pictures demonstrate the wide applicability of the algorithm as well as the beauty of minimal surfaces.

##### MSC:
 65D17 Computer aided design (modeling of curves and surfaces) 53A10 Minimal surfaces, surfaces with prescribed mean curvature