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.

We study the evolution of complete non-compact convex hypersurfaces in R n+1 by the inverse mean curvature flow. We establish the long time existence of solutions and provide the characterization of the maximal time of existence in terms of the tangent cone at infinity of the initial hypersurface. Our proof is based on an a'priori pointwise estimate on the mean curvature of the solution from below in terms of the aperture of a supporting cone at infinity. The strict convexity of convex solutions is shown by means of viscosity solutions. Our methods also give an alternative proof of the result by Huisken and Ilmanen in [24] on compact start-shaped solutions, based on maximum principle argument.

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This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in R 2 under the α-curve shortening flow for exponents α > 1 2 . We show that any such curve having in addition its two ends asymptotic to two parallel lines, converges under α-curve shortening flow to the unique translating soliton whose ends are asymptotic to the same parallel lines. This is a new result even in the standard case α = 1, and we prove for all exponents up to the critical case α > 1 2 .

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