Convergence of the fractional Yamabe flow for a class of initial data

TL;DR

This paper proves the convergence of the fractional Yamabe flow for certain initial data classes, extending previous existence results and utilizing simplified approaches from Struwe's work.

Contribution

It establishes convergence of the fractional Yamabe flow for a broader class of initial data, advancing understanding of fractional geometric flows.

Findings

Proved convergence of the fractional Yamabe flow for specific initial data.

Extended previous existence results to convergence results.

Utilized simplified methods from Struwe's approach.

Abstract

This work is a follow-up on the work of the second author with P. Daskalopoulos and J.L. V\'{a}zquez. In this latter work, we introduced the Yamabe flow associated to the so-called fractional curvature and prove some existence result of mild (semi-group) solutions. In the present work, we continue this study by proving that for some class of data one can prove actually convergence of the flow in a more general context. We build on the approach simplified in the book of M. Struwe.