In this talk, I will describe joint work with Charles Baker. We will consider surfaces of co-dimension two in Euclidean space moving by the mean curvature flow. We show that if the initial surface satisfies a nonlinear curvature condition depending on the normal curvature tensor then the mean curvature flow deforms the surface to a round point.

Given a prescription of paths on a surface — one for every direction in each tangent space — one might ask if those paths are the geodesics of a Riemannian metric. Generically they are not, hence one might look for a Riemannian metric whose geodesics are ‘as close as possible’ to the prescribed paths. This gives rise to a natural variational problem. In this talk I will discuss how its critical points relate to certain weakly conformal maps and complex geometry.