Type II Singularities on complete non-compact Yamabe flow

TL;DR

This paper analyzes Type II singularities in the Yamabe flow on non-compact manifolds, identifying blow-up rates and showing convergence to a steady soliton, revealing new insights into singularity models.

Contribution

It demonstrates for the first time that a steady soliton can serve as a finite-time singularity model in the Yamabe flow.

Findings

Determined the blow-up rate of maximum curvature.

Proved convergence to a steady soliton after rescaling.

Identified the steady soliton as a finite-time singularity model.

Abstract

This work concerns with the existence and detailed asymptotic analysis of Type II singularities for solutions to complete non-compact conformally flat Yamabe flow with cylindrical behavior at infinity. We provide the specific blow-up rate of the maximum curvature and show that the solution converges, after blowing-up around the curvature maximum points, to a rotationally symmetric steady soliton. It is the first time that the steady soliton is shown to be a finite time singularity model of the Yamabe flow.