Sfb 288 Differential Geometry and Quantum Physics

Affine Spheres
The DPW-method applied to Affine Spheres

Initally the Dorfmeister-Pedit-Wu-method was applied to CMC surfaces. In particular this gives an algorithm to compute all CMC surfaces. Moreover in this method it is possible to add umbilics to the date of such a surface.
However, the method had been applied later to other problems in differential geometry. Also in discrete differential geometry it shows to be a useful method. In a recent work, Dorfmeister and Eitner have applied to method to another kind of surfaces, affine spheres, i.e. surfaces immersed into the 3-dimensional (equi)-affine space which have the property, that their affine normal vector field intersects at one finite point.
What is the advantage of the method? The problem of solving a partial differential equation first (the compatibility condition) and then the linear system of the moving frame is here reduced to solve two ordinary linear differential equations and solve some algebraical equation. However, the even if the first step might not cause problems, the second step is - at least if one wants to find the solution analytical - rather complicated. On the other side the method is very simple to program, and therefore always useful for finding numerical solutions of the grounding geometrical problem.

Even though the method seems very simple, at least for numerical approach, one has to be very careful. The ordinary differential equations have to be solved in a very particular Lie-loop-group. To do so, one have to find integrator matrices in the corresponding Lie-loop-algebra. Moerover one have to ensure that the integration method is consistent and stable. The most simple method, the explicit Euler-method is not stable. However, the integrators of this method give the transport-matrices in the discrete model of affine sphere discovered by Bobenko and Schief.

A picture of what is not an affine sphere

The DPW shows that any negative curved affine sphere is determined by two asymtotical lines in space. The potentials (they correspondence to the meromorphic potential in the CMC case has the same kind of entries: the Blaschke-metric initially along the two asymtotical lines and the coefficients of the cubic form.

Using in implicit Euler method in the loop-algebra

Author: Ulrich Eitner