Sfb 288 Differential Geometry and Quantum Physics

Discrete Surfaces
Discrete CMC Surfaces by the DPW-Method

The discrete DPW method is a recipe for generating discrete surfaces of constant mean curvature (cmc) out of (discrete) holomorphic data. Given a discrete holomorphic map (on the left side one sees a discrete z and a discrete Exp(z)) one has to:

• applesSolve a discrete partial difference equation of the form
  G_n+1,m= U_n,m G_n,m
  G_n,m+1= V_n,m G_n,m
  

  Where U and V are lambda-dependent matrices. They are determined by the holomorphic map.

• Do some splitting in the corresponding loop group.
• Use the Sym formula to obtain the discrete cmc surface

This method is quite efficient since the splitting can be done exactly (in the continious case one has to approximate it numerically). Moreover it is possible to construct examples with umbillics (the three legged Mr Bubble on the left side is one. The umbillic is on top of his head.)

discrete z2/3	discrete exp(z)
Above holomorphic maps will give the surfaces below.

Author: Tim Hoffmann