Counterexample to the Maximum Principle of Discrete Minimal Surfaces

By Konrad Polthier and Wayne Rossman

The maximum principle for solutions of elliptic partial differential equations says, that a solution cannot lie on one side of another solution unless both solutions are identical. This maximum principle does not hold for discrete minimal surfaces as the folowing example demonstrates: if the central vertex lies outside the convex hull of its boundary then any xy-planar triangulation, which is certainly discrete minimal, lies one side of the example. For example, consider the planar triangulation obtained from projecting the surface onto the xy-plane. The projection is combinatorically equivalent, and the surface is a graph over the planar projection.

Note, since the identity map of a discrete minimal surface is a discrete harmonic map, this example also demonstrates that the mean value property and convex hull property of discrete harmonic maps do not hold. Further note, that both properties hold in special situations where either all triangles are equilateral or have all vertex angles in [-90,90] degrees. In this example, the center vertex lies on the convex hull exactly at u = 2 which is the situation when the first vertex angle becomes 90 degrees. Increasing u further leads to an increasing angle. To better understand the influence of the vertex angles have a look at the paper Pinkall/Polthier with an explicit formula of the balancing formula for discrete minimal surfaces.

Note, that the discrete maximum principle does hold for the five-vertex Laplacian defined over the special rectangular ZxZ grid.

Usage of the Applet

Slider 'u'	The parameter 'u' determines the boundary shape. It provides the position of the two opposite vertices (+u,0,-u) and (-u,0,-u). During area minimization the boundary vertices are fixed while the central vertex is free to move in space. The final position of the central vertex after pressing the 'Minimize' button minimizes the area of the polyhedral surface. For u in [0,2] the central vertex lies within the convex hull of the boundary after minimization. The remarkable fact is that this property does not hold for u > 2 when the minimum position of the central vertex is outside the convex hull of the boundary.
Slider 'h'	The parameter 'h' determines the height of the center vertex. Vary this value and observe how the area of the configuration changes. After pressing the button "Minimize" the value is set to the height value which minimizes the area of the polyhedral surface.
Vertices and Faces	The parameter 'u' is controlled by the slider. The vertices are indexed starting at 0 to numVertices-1. The face list references to the indexed vertices. <points> <pf>-u 0. -u </pf> <pf> u 0. -u </pf> <pf>-1. 1. 0.</pf> <pf> 1. 1. 0.</pf> <pf>-1. -1. 0.</pf> <pf> 1. -1. 0.</pf> <pf> 0. 0. h </pf> </points> <faces> <f>0 6 2</f> <f>6 3 2</f> <f>6 1 3</f> <f>0 4 6</f> <f>4 5 6</f> <f>5 1 6</f> </faces>

This applet exists as an attachment to the model http://www.eg-models.de/2000.11.040 published at the Electronic Geometry Models server.