We study the extinction behaviour of solutions to the fast diffusion equation ut = Δum on ℝN × (0, T), in the range of exponents . We show that if the initial value u0 is trapped in between two Barenblatt solutions vanishing at time T, then the vanishing behaviour of u at T is given by a Barenblatt solution. We also give an example showing that for such a behaviour the bound from above by a Barenblatt solution B (vanishing at T) is crucial: we construct a class of solutions u with initial value u0 = B(1 + o(1)), near |x| » 1, which live longer than B and change behaviour at T. The behaviour of such solutions is governed by B(·, t) up to T, while for t > T the solutions become integrable and exhibit a different vanishing profile. For the Yamabe flow the above means that these solutions u develop a singularity at time T, when the Barenblatt solution disappears, and at t > T they immediately smoothen up and exhibit the vanishing profile of a sphere.
We construct new type II ancient compact solutions to the Yamabe flow. Our solutions
are rotationally symmetric and converge, as , to a tower of two spheres.
Their curvature operator changes sign.
We allow two time-dependent parameters in our ansatz. We use perturbation theory, via fixed point arguments,
based on sharp estimates on ancient solutions of the approximated linear equation
and careful estimation of the error terms which allow us to make the right choice of parameters.
Our technique may be viewed as a parabolic analogue of gluing two exact solutions to the rescaled equation, that is the spheres, with narrow cylindrical necks to obtain a new ancient solution to the Yamabe flow. The result
generalizes to the gluing of k spheres for any , in such a way the configuration of radii of the spheres glued is driven as by a First order Toda system.
As part of the general investigation of Ricci flow on complete surfaces with finite total curvature, we study this flow for surfaces with asymptotically conical (which includes as a special case asymptotically Euclidean) geometries. After establishing long-time existence, and in particular the fact that the flow preserves the asymptotically conic geometry, we prove that the solution metric g(t) expands at a locally uniform linear rate; moreover, the rescaled family of metrics t−1g(t) exhibits a transition at infinite time inasmuch as it converges locally uniformly to a complete, finite area hyperbolic metric which is the unique uniformizing metric in the conformal class of the initial metric g0.