The Weierstrass-Enneper Parameterization for minimal surfaces gives minimal surfaces in terms of complex holomorphic functions. The vanishing condition of gradient of Dirichlet functional gives algebraic constraints on interior control points of a Bezier surface yielding a quasi-minimal surface. For these quasi-minimal surfaces the mean curvature is not zero and they not isothermal. Can we talk about implication of Weierstrass Enneper representation for such quasi-minimal surfaces?
Dear Ahmad,
I think that Weierstrass-Enneper representation always gives the surface which the mean curvature vanishing.
> Can we talk about implication of Weierstrass Enneper representation for such quasi-minimal surfaces?
This will be interesting !
I wil think about it.
The natural generalisation of the Weierstrass representation of minimal surfaces is only possible in the context of spin geometry, as studied by many authors since approx. 1998. The idea is to start with a manifold with a parallel spinor (for example, on flat space), and to restrict it to a submanifold. The spinor then satisfies an equation D\psi = H\psi, H being the second fundamental form. H=0 is then just a very special case. For example, the Lorentzian setting that you need is explained in the paper
Immersions of Lorentzian surfaces in R^{2,1}
by Marie-Amélie Lawn; J. Geom.Phys.2008. From there, the references will guide you through the extensive literature.
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