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

TL;DR

This paper studies the formation of singularities in complete non-compact solutions to the Yamabe flow, revealing their profiles as Yamabe solitons and classifying their extinction behaviors.

Contribution

It characterizes the singularity profiles of Yamabe flow solutions with cylindrical behavior at infinity, linking them to Yamabe solitons and describing their extinction and long-term behaviors.

Findings

Singularity profiles are Yamabe solitons determined by initial data asymptotics.

Solutions may become extinct at time T or persist longer, with profiles described by shrinkers and expanders.

The asymptotic behavior at infinity influences the type of singularity and the profile shape.

Abstract

This work addresses the {\em singularity formation} of complete non-compact solutions to the conformally flat Yamabe flow whose conformal factors have {\em cylindrical behavior at infinity}. Their singularity profiles happen to be {\em Yamabe solitons}, which are {\em 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 {\em Yamabe shrinker} that becomes extinct at time $T$. In the second case, the singularity profile is described by a {\em singular} Yamabe shrinker slightly before $T$ and by a matching {\em Yamabe expander} slightly after $T$ .