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

This is joint work with Eric Chen and Gilles Carron.