Contents Online

# Communications in Analysis and Geometry

## Volume 27 (2019)

### Number 8

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

Pages: 1757 – 1798

DOI: https://dx.doi.org/10.4310/CAG.2019.v27.n8.a4

#### Authors

#### Abstract

This work addresses the *singularity formation* of complete noncompact 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 $T$.

Received 20 December 2013

Accepted 3 May 2016

Published 21 January 2020