  This paper quantifies the speed of convergence and higher- order asymptotics
of fast diffusion dynamics on R^n to the Barenblatt (self similar) solution.
Degeneracies in the parabolicity of this equation are cured by re-expressing
the dynamics on a manifold with a cylindrical end, called the cigar. The
nonlinear evolution becomes differentiable in Hoelder spaces on the cigar. The
linearization of the dynamics is given by the Laplace-Beltrami operator plus a
transport term (which can be suppressed by introducing appropriate weights into
the function space norm), plus a finite-depth potential well with a universal
profile. In the limiting case of the (linear) heat equation, the depth
diverges, the number of eigenstates increases without bound, and the continuous
spectrum recedes to infinity. We provide a detailed study of the linear and
nonlinear problems in Hoelder spaces on the cigar, including a sharp
boundedness estimate for the semi-group, and use this as a tool to obtain sharp
convergence results toward the Barenblatt solution, and higher order
asymptotics. In finer convergence results (after modding out symmetries of the
problem), a subtle interplay between convergence rates and tail behavior is
revealed. The difficulties involved in choosing the right functional spaces in
which to carry out the analysis can be interpreted as genuine features of the
equation rather than mere annoying technicalities.
