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Deforming C2 with a pair of holomorphic foliations

I went over my old notes on deforming C2. Pairs of laminations of C2 by holomorphic curves was a recurring theme in holomorphic dynamics, and I worked out the conditions using the Neienhaus tensor for doing quasiconformal deformation on a pair of laminations and obtaining a new complex structure on C2. The point is more or less that two transverse laminations are deformed into two different transverse laminations, which goes along with the leaves of each of those laminations undergoing a quasiconformal deformation. I had worked out a few geometric basics as well, sort of to check the viability of the idea. For smooth deformations locally one would have an R linear map of C2. It turns out that there are two open conditions for such maps - they either map two complex lines to two complex lines or else they map no complex lines to complex lines. I don't think this is a direction to pursue right now, as the condition one obtains on the beltrami coefficients is simple and elegant, yet it is a nonlinear PDE. Given the implicit geometric meaning it might be interesting in looking at whether the property of being less than norm 1 propagates in a solution.

I need to rewrite the equations using bi=fzbarfz since bi IS a beltrami coefficient.

There was some horrible thing that happened when you tried to develop a series solution, and I don't recall what it was.

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