The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant

We study the Yamabe flow on asymptotically flat manifolds with non-positive Yamabe constant Y < 0. Previous work by the second and third named authors [10] showed that while the Yamabe flow always converges in a global weighted sense when Y > 0, the flow must diverge when Y < 0. We show here in the Y < 0 case however that after suitable rescalings, the Yamabe flow starting from any asymptotically flat manifold must converge to the unique positive function which solves the Yamabe problem on a compactification of the original manifold.(c) 2022 Published by Elsevier Inc.