I went over my old notes on deforming $\mathbb{C}^2$. Pairs of laminations of $\mathbb{C}^2$ 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 $\mathbb{C}^2$. 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 $\mathbb{R}$ linear map of $\mathbb{C}^2$. 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 \[b_i = \dfrac{\frac{f}{zbar}}{\frac{f}{z}}\] since $b_i$ 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.
No comments:
Post a Comment