PDE Seminar Abstracts

A tale of two geometric biharmonic heat flows

Glen Wheeler
University of Wollongong, Australia
Monday 31 August 2015, 2-3pm, Carslaw Room 829 (AGR)


For a smoothly immersed submanifold, the Laplacian of the immersion is the mean curvature. It is therefore unambiguous to talk about the geometric heat flow of immersions: this is the mean curvature flow. For the iterated Laplacian, two perspectives collide: first, we may take the point of view that the mean curvature vector is a section of the normal bundle, and apply the induced normal Laplacian to the mean curvature vector. This gives us the surface diffusion operator. Second, there is the perspective that the mean curvature vector lives in Euclidean space, and so it may be acted upon by the rough Laplacian, the same operator that acts on the immersion to produce the mean curvature vector. If we take this perspective we uncover a new curvature flow, which we call the Chen flow. In this talk we describe recent work on the surface diffusion flow and the Chen flow, highlighting the (dramatic) differences between the two.