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  • Author: Panagiota Daskalopoulos x
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Abstract

Ancient solutions play an important role in studying singularities. These are special solutions to an evolution equation that exists for all time −∞ < t ≤ T, with T ≤ +∞. They typically appear as blow-up limits near a singularity. We will discuss some of the recent developments regarding the classification of ancient solutions to geometric flows, in particular the mean curvature flow and the Ricci flow.

Abstract

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 u 0 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 u 0 = 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.

Abstract

We construct new type II ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as t-, 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 k2, in such a way the configuration of radii of the spheres glued is driven as t- by a First order Toda system.

Abstract

This work concerns with the existence and detailed asymptotic analysis of type II singularities for solutions to complete non-compact conformally flat Yamabe flow with cylindrical behavior at infinity. We provide the specific blow-up rate of the maximum curvature and show that the solution converges, after blowing-up around the curvature maximum points, to a rotationally symmetric steady soliton. It is the first time that the steady soliton is shown to be a finite time singularity model of the Yamabe flow.

Abstract

We prove the all-time existence of non-compact, complete, strictly convex solutions to the α-Gauss curvature flow for any positive power α.