Classification of singularities in the complete conformally flat Yamabe flow

TL;DR

This paper classifies singularities in the complete conformally flat Yamabe flow, showing that under certain conditions, the flow's singularity at finite time is always of type I, and characterizes eternal solutions as gradient steady solitons.

Contribution

It provides a classification of singularities in the Yamabe flow and characterizes eternal solutions as gradient steady solitons under specific curvature conditions.

Findings

Eternal solutions with bounded scalar curvature are gradient steady solitons.

Flow singularities at finite time are always of type I.

Under certain conditions, the flow's blow-up behavior is well-understood.

Abstract

We show that an eternal solution to a complete, locally conformally flat Yamabe flow, $\frac{\partial}{\partial t} g = -Rg$, with uniformly bounded scalar curvature and positive Ricci curvature at $t = 0$, where the scalar curvature assumes its maximum is a gradient steady soliton. As an application of that, we study the blow up behavior of $g(t)$ at the maximal time of existence, $T < \infty$. We assume that $(M,g(\cdot, t))$ satisfies (i) the injectivity radius bound {\bf or} (ii) the Schouten tensor is positive at time $t = 0$ and the scalar curvature bounded at each time-slice. We show that the singularity the flow develops at time $T$ is always of type I.