Vortex distribution in the lowest Landau level

We study the vortex distribution of the wave functions minimizing the Gross-Pitaevskii energy for a fast rotating condensate in the lowest Landau level (LLL): we prove that the minimizer cannot have a finite number of zeroes, thus the lattice is infinite, but not uniform. This uses the explicit expression of the projector onto the LLL. We also show that any slow varying envelope function can be approximated in the LLL by distorting the lattice. This is used in particular to approximate the inverted parabola and understand the role of "invisible" vortices: the distortion of the lattice is very small in the Thomas-Fermi region but quite large outside, where the "invisible" vortices lie.