Communications in Analysis and Geometry

Volume 27 (2019)

Number 8

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

Pages: 1757 – 1798



Panagiota Daskalopoulos (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

John King (School of Mathematical Sciences, University of Nottingham, United Kingdom)

Natasa Sesum (Department of Mathematics, Rutgers University, Piscataway, New Jersey, U.S.A.)


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