previous to Discrete Conformal Maps as integrable systems

The spectral curve for for a Discrete Conformal Map

Let tex2html_wrap_inline100 be the ordered points tex2html_wrap_inline102 . We think of this as the initial discrete curve. If we choose any point tex2html_wrap_inline104 on tex2html_wrap_inline76 not on tex2html_wrap_inline100 then for fixed cross-ratio q there is a unique point tex2html_wrap_inline112 for which


Inductively we define tex2html_wrap_inline114 by requiring


Clearly these points are given by iterating the Mobius transformations tex2html_wrap_inline116 uniquely determined by the condition that for all points tex2html_wrap_inline118


Thus tex2html_wrap_inline100 generates another discrete curve tex2html_wrap_inline122 . This will also be periodic of period n when tex2html_wrap_inline126 . Since tex2html_wrap_inline128 this means the first point tex2html_wrap_inline104 must be an eigenline of the "holonomy"


It is easy to show that the transformation tex2html_wrap_inline132 can be represented by a tex2html_wrap_inline134 matrix of the form tex2html_wrap_inline136 where tex2html_wrap_inline138 is a projection matrix. Therefore if we allow q to vary over all complex values (and rename it tex2html_wrap_inline142 ) we can represent the family of all holonomies by a matrix valued polynomial tex2html_wrap_inline144 in tex2html_wrap_inline146 . The spectral curve is essentially the characteristic polynomial of this matrix. The important point, which is not obvious, is that this curve is independent of the point tex2html_wrap_inline148 at which the holonomy is based (it can be shown that a change of base point amounts to a conjugation of the holonomy matrix, hence its charateristic polynomial remains invariant).

Each point tex2html_wrap_inline148 itself is an eigenline of the holonomy matrix tex2html_wrap_inline152 for tex2html_wrap_inline154 . The family of all eigenlines for fixed k,m as tex2html_wrap_inline142 varies gives a line bundle over the spectral curve. This line bundle changes with k and m in an essentially elementary way: it is determined by a map from tex2html_wrap_inline164 into the Jacobi variety of the spectral curve which is either a group homomorphism or a zigzag (our term for a map which is only a homomorphism on the subgroup of pairs (k,m) for which k+m is even). We can provide a reconstruction of the discrete conformal map using Riemann tex2html_wrap_inline98 -functions to construct the appropriate sections of the moving line bundle: the formulae are very like the well-known formulae for Baker-Akhiezer functions in soliton theory, but the discrete versions have only poles instead of essential singularities.

There is an elegant geometric connection between these discrete maps and their smooth cousins in soliton theory. In the sense of Capel and Nijhoff the constant cross-ratio equation is the discrete analogue of the Schwarzian KdV equation


where tex2html_wrap_inline172 is the Schwarzian derivative of z. This equation is solvable by linearising the flow on the Jacobi variety of a curve (Riemann surface). The line followed by the x-flow is tangent to the Abel image of the curve at a base point which must be an order two branch point for the curve, while the t-flow follows the third order osculating line to this point. The discrete dynamics are obtained by replacing each of these with a secant to the curve: the location of the other end of each secant determines the cross-ratio of the discrete conformal map. It is interesting to note that the zigzag (which is the generic type of solution) arises from splitting the base point into two (unbranched) points. This phenonmenon has no analogue in the smooth theory.

More pictures:

previous to Discrete Conformal Maps as integrable systems

Ian McIntosh
Wed May 13 14:19:03 EDT 1998