Extinction profile of complete non-compact solutions to the Yamabe flow

This work addresses the { singularity formation} of complete non-compact solutions to the conformally flat Yamabe flow whose conformal factors have { cylindrical behavior at infinity}. Their singularity profiles happen to be { Yamabe solitons}, which are { self-similar solutions} to the fast diffusion equation satisfied by the conformal factor of the evolving metric. The self-similar profile is determined by the second order asymptotics at infinity of the initial data which is matched with that of the corresponding self-similar solution. Solutions may become extinct at the extinction time T of the cylindrical tail or may live longer than T. In the first case the singularity profile is described by a { Yamabe shrinker} that becomes extinct at time T. In the second case, the singularity profile is described by a { singular} Yamabe shrinker slightly before T and by a matching { Yamabe expander} slightly after T .