In this talk, we will discuss the behavior of the Yamabe flow on an asymptotically flat (AF) manifold. We will first show the long-time existence of the Yamabe flow starting from an AF manifold and discuss the uniform estimates on manifolds with positive Yamabe constant. This would allow us to prove global weighted convergence along the Yamabe flow on such manifolds. We also prove that the flow will diverge if the Yamabe constant is nonpositive. It turns out that the blowup behavior of the flow starting from manifolds with nonpositive Yamabe constant is explicit. We will discuss some further work in this direction.

This is joint work with Eric Chen and Gilles Carron.