Discrete constant mean curvature surfaces and their index

Abstract

We define triangulated piecewise linear constant mean curvature surfaces using a variational characterization. These surfaces are critical for area amongst continuous piecewise linear variations which preserve the boundary conditions, the simplicial structures, and (in the nonminimal case) the volume to one side of the surfaces. We then find explicit formulas for complete examples, such as discrete minimal catenoids and helicoids. We use these discrete surfaces to study the index of unstable minimal surfaces, by numerically evaluating the spectra of their Jacobi operators. Our numerical estimates confirm known results on the index of some smooth minimal surfaces, and provide additional information regarding their area-reducing variations. The approach here deviates from other numerical investigations in that we add geometric interpretation to the discrete surfaces.