We know that every surface of genus ($g\geq 2$) admits a pair of pants decomposition. And there is the Fenchel Nielsen Coordinates on the Teichmuller space associated to such a decomposition where we have the length functions of the geodesics boundaries and the twisting parameters for gluing these boundaries.
My question is the following:
We choose again a pair of pants decomposition. Then there arises the trivalent graph representing this decomposition.
We could associate a fixed a pair of pants to each of vertices, say of boundary geodesic length (1,1,1). And then glue flat cylinders of both boundary length 1 and height $h$ to form a Riemann surface(of course we will also need the twisting parameter for the sewing, but it seems that we should only need the relative twisting parameter as cylinder itself has a rotation symmetry.).
The height of these cylinders and the relative twisting parameter together form 6g-6 parameters. Will they also be a parametrization of Teichmuller space?