Rasmus Johansen Jouttijärvi: Boundary conditions for Ricci flow

If we want to deform a compact Riemannian manifold with boundary using Ricci flow, we first need to decide on appropriate boundary conditions. It is natural to impose the Dirichlet-like condition that the conformal class of the metric on the boundary is preserved, and a Neumann condition on the second fundamental form of the boundary.   An appropriate choice of condition on the second fundamental form does not seem straightforward and several alternatives have been studied in the past. One such choice, is the condition that the boundary be umbilic (i.e. that the second fundamental form is a constant multiple of the metric), another choice could be that the boundary is minimal (i.e. that the second fundamental form is trace-free). In this talk, we propose another alternative, in the form of a condition on the evolution of the trace of the second fundamental form (i.e. the mean curvature). The condition in question is derived from the study of Perelman's lambda-functional, which is closely related to Ricci flow.   We provide an overview of the method with which to prove short time existence of solutions to the Ricci flow initial value problem, under the aforementioned boundary conditions.