Two stable and an unstable discrete minimal graph over the same planar domain with the same boundary and mesh.
We define discrete compact minimal surfaces as piecewise linear continuous compact triangulated surfaces that are critical for the area functional with respect to all variations through surfaces of the same type that preserve the simplicial structure and the boundary condition (see  and model  on this server).
Although discrete minimal surfaces are critical for area, they are not necessarily area minimizing. Shown here are three discrete minimal surfaces  that are annular graphs over the same domain and posess the same boundary (the boundary has two components). The first and third surface is area minimizing, but the second surface is unstable and not area minimizing. In the unstable surface we include variation vectors (in yellow) at the interior vertices of the surface showing how the surface can be varied so that area decreases in second order.
The examples here are of interest because they show how discrete minimal surfaces can have properties which differ from those of smooth minimal surfaces. For example, any smooth minimal graph is uniquely determined by its boundary. Also, any smooth minimal graph is stable, see the work of Barbosa and do Carmo . The examples here show that neither of these properties hold for discrete minimal surfaces.
In contrast to the existence of these counterexamples, we conjecture that the above properties of smooth minimal graphs remain true for discrete minimal graphs over simply connected convex domains. (Note that the two examples here are graphs over annular domains.)
Model produced with: JavaView version 2.00.a2
|Keywords||Minimal Surface; Discrete Surface; Graph; Stability|
|MSC-2000 Classification||53-04 (53-XX, 68Uxx, 68Rxx, 65Kxx, 65M50)|
Submitted: Mon Jan 22 09:21:14 CET 2001.
Revised: Fri Jul 20 11:03:03 CET 2001, Mon Aug 27 22:10:02 CET 2001.
Accepted: Tue Sep 11 15:18:14 CET 2001.
Technische Universität BerlinWayne Rossman
Straße des 17. Juni 136
Faculty of Science
Rokko, Kobe 657-8501