Singularities of Discrete Indefinite Affine Minimal Surfaces

Abstract

A smooth affine minimal surface with indefinite metric can be obtained from a pair of smooth non-intersecting spatial curves by Lelieuvre's formulas. These surfaces may present singularities, which are generically cuspidal edges and swallowtails. By discretizing the initial curves, one can obtain by the discrete Lelieuvre's formulas a discrete affine minimal surface with indefinite metric. The aim of this paper is to define the singular edges and vertices of the corresponding discrete asymptotic net in such a way that the most relevant properties of the singular set of the smooth version remain valid.

Figures (4)

• Figure 1: Forbidden configuration of the projections of $\alpha$ and $\beta$ in $\pi$, assuming that $(u_0,v_0+\tfrac{1}{2})$ is a singular edge.
• Figure 2: Possible configurations of the projections of $\alpha$ and $\beta$ in $\pi$, assuming that $(u_0,v_0+\tfrac{1}{2})$ is a singular edge. In the up-left, up-right and down figures, $P\beta(v-1)$ is positioned in quadrant A, B or C, respectively.
• Figure 3: Star configurations of cases A (left), B (middle) and C (right). Thick full segments correspond to singular edges.
• Figure 4: Example \ref{['ex:1']} with three different values for $y$, corresponding to configurations A, B and C. The different colors correspond to the bilinear interpolations at the four quadrangles.

Theorems & Definitions (13)

• Proposition 2.1
• proof
• Proposition 2.2
• Proposition 2.3
• proof
• Proposition 4.1
• proof
• Proposition 4.2
• proof
• Proposition 4.3