Yamabe Flow: steady solitons and type II singularities

TL;DR

This paper investigates the behavior of the Yamabe flow on non-compact conformally flat manifolds, focusing on convergence to steady solitons and the formation of Type II singularities at finite or infinite times.

Contribution

It establishes conditions for convergence to Yamabe steady solitons and proves the existence of Type II singularities in the flow.

Findings

Complete non-compact conformally flat solutions converge to steady solitons.

Type II singularities can develop at finite or infinite times.

The paper characterizes the circumstances under which singularities occur.

Abstract

We study the convergence of complete non-compact conformally flat solutions to the Yamabe flow to Yamabe steady solitons. We also prove the existence of Type II singularities which develop at either a finite time $T$ or as $t \to +\infty$.