  The renormalization group transformation for extreme value statistics of
independent, identically distributed variables, recently introduced to describe
finite size effects, is presented here in terms of a partial differential
equation (PDE). This yields a flow in function space and gives rise to the
known family of Fisher-Tippett limit distributions as fixed points, together
with the universal eigenfunctions around them. The PDE turns out to handle
correctly distributions even having discontinuities. Remarkably, the PDE admits
exact solutions in terms of eigenfunctions even farther from the fixed points.
In particular, such are unstable manifolds emanating from and returning to the
Gumbel fixed point, when the running eigenvalue and the perturbation strength
parameter obey a pair of coupled ordinary differential equations. Exact
renormalization trajectories corresponding to linear combinations of
eigenfunctions can also be given, and it is shown that such are all solutions
of the PDE. Explicit formulas for some invariant manifolds in the Fr\'echet and
Weibull cases are also presented. Finally, the similarity between
renormalization flows for extreme value statistics and the central limit
problem is stressed, whence follows the equivalence of the formulas for Weibull
distributions and the moment generating function of symmetric L\'evy stable
distributions.
