Speaker: Sathya Rengaswami

Title: Some Radially Symmetric Solutions to Curvature Flows

Abstract:  Analogous to the bowl soliton of mean curvature flow, we construct rotationally symmetric translating solutions to a large class of extrinsic curvature flows, namely those whose speeds are 𝛼-homogeneous (𝛼>0), elliptic and symmetric with respect to the principal curvatures. We show that these solutions are necessarily convex, and give criteria for the speed functions which determine whether these translators are defined over all of \mathbf{R}^{n} or contained in a cylinder. For speeds that are nonzero when at least one of the principal curvatures is nonzero, we are also able to describe the asymptotics of the translator at infinity.

More specifically in the case of 1-homogeneous speeds, we discuss the question of existence and uniqueness of pancake solutions, which are collapsed radially symmetric ancient solutions that lie in a slab. We also describe their asymptotics as t →-∞.