2019 Theses Doctoral

Non-compact geometric flows: long time existence and type II singularities

Choi, Beomjun

In this work, we study how solutions of certain non-compact geometric flows of fast-diffusion type interact with their asymptotic geometries at infinity. In the first part, we show the long time existence theorem to the inverse mean curvature flow for complete convex non-compact initial hypersurfaces. The existence and behavior of a solution is tied with the evolution of its tangent cone at infinity. In particular, the maximal time of existence can be written in terms of the area ratio between the initial tangent cone at infinity and the flat hyperplane. In the second part, we study the formation of type II singularity for non-compact Yamabe flow. Assuming the initial metric is conformally flat and asymptotic to a cylinder, we show the higher order asymptotics of the metric determines the curvature blow-up rates at the tip in its first singular time. We also show the singularities of such solutions are modeled on rotationally symmetric steady gradient solitons.