Spectral curves of polygons and triangulated tori
Project leader: U. Pinkall
We investigate an approach to discrete conformality based on the
notion of holomorphic line bundles over "discrete surfaces", that
is, over vertex sets of triangulated surfaces with black and white
colored faces. As a special case, we give a reinterpretation of
Dynnikov's and Novikov's approach to conformal maps to
S^2=CP^1 which reveals it as the first example of a theory
of discrete holomorphicity that is at the same time Möbius-invariant
and governed by linear equations.
We introduce Darboux transformations for arbitrary immersions of
discrete surfaces into S^4=HP^1 which can be interpreted as
a time discrete Davey-Stewartson flow on the space of immersions. For
generic immersions of discrete tori with regular combinatorics, we
show that the space of Darboux transformations can be desingularized
to a compact Riemann surface (the spectral curve) thus making
available powerful methods from the theory of algebraically completely
integrable systems.
In the second period, beyond the soliton theory of triangulated
surfaces, our investigations will concentrate on developing a
definition of conformality for immersions of "discrete Riemann
surfaces". Moreover, we plan to study a new class of "discrete
minimal surfaces" that appears naturally in the context of our
investigations.
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