Deforming complex space using Beltrami coefficients on a pair of foliations

Since the pair of equations for deforming a pair of foliations is a pair of PDSs involving the beltrami coefficients, I checked to see whether they were simpler to work with if the beltrami coefficients were in turn represented as a ratio involving $\partial \bar{z}$ and $\partial z$ but that didn’t help.

I had worked out a series solution to them years ago, looking for simple solutions. The simple solutions all looked like they turned out to be trivial as I recall (i.e. no different than a deformation of each coordinate separately).

However, I went ahead and wrote out the series solution again just because the PDEs involved obviously have a series solution, and PDE’s can be hard to solve.

If $b_1$ and $b_2$ are the beltrami coefficients of the horizontal and vertical holomorphic leaves, the PDE pair is

\[\dfrac{\partial b_1}{\partial \bar{z}_2} = b_2 \dfrac{\partial b_1}{\partial z_2}\]

and

\[\dfrac{\partial b_2}{\partial \bar{z}_1} = b_1 \dfrac{\partial b_2}{\partial z_1}\]

You can select any two convergent series \[\sum a_{ijk} z_1^i z_2^j \bar{z}_1^k\] and \[\sum c_{ijk} z_1^i z_2^j \bar{z}_2^k\] then there is a nice series relationship that gives you actual solutions $b_1$ and $b_2$ as an infinite series. The series solution is nicer if you actually represent $b_1$ and $b_2$ as
\[ b_1 = \sum A_{ij} \bar{z}_1^i \bar{z}_2^j\]
and
\[ b_2 = \sum C_{ij} \bar{z}_1^i \bar{z}_2^j\]

where $A_{ij}$ and $C_{ij}$ are holomorphic functions. 

You then get the relationships:

\[ (j+1) A_{i,j+1} = \sum_{k \leq i, \ell \leq j} \dfrac{\partial A_{k\ell}}{\partial z_2} C_{i-k,j-\ell}\]

\[ (i+1) C_{i + 1,j} = \sum_{k \leq i, \ell \leq j} \dfrac{\partial C_{k\ell}}{\partial z_1} A_{i-k,j-\ell}\]

This recursive relationship requires that you specify each $A_{i0}$ and each $C_{0j}$ as a holomorphic function.

Of course, where resulting series converges is a different matter altogether. But it appears this gives lots of solutions to the system of PDEs. For example, if the constants $a_{ijk}$ and $c_{ijk}$ get small very rapidly, this method would give you convergent solutions to the PDE.

Now, of course, there are many solutions, because one can simply take two transverse holomorphic motions and turn them into a deformation by treating them as a coordinate system. That is, take two different pairs of transverse foliations. For each foliation in each pair choose a leaf. Choose a way to match the selected two leaves that came from one pair with the selected two leaves that came from the other pair. Now extend by treating the transverse foliations as giving coordinates.

The PDE is written for the case where one of these pairs is simply the regular horizontal and vertical leaves, and we are specifying beltrami coefficients to deform the structure of those leaves.

So there are lots of solutions. But it is interesting to have solutions specified by beltrami coefficients. It means you can deform an existing structure and still have a holomorphic space.