Fast diffusion flow on manifolds of nonpositive curvature

We consider the fast diffusion equation on a nonparabolic Riemannian manifold M. Existence of weak solutions holds. Then we show that the validity of Euclidean–type Sobolev inequalities implies that smoothing effects into L^\infty hold true. The converse holds if m is sufficiently close to one. We then consider the case in which the manifold has the addition gap property min σ(−\Delta) > 0, \Delta being the Laplace-Beltrami operator. In that case solutions vanish in finite time, and we estimate from below and from above the extinction time.