Convergence of the fractional Yamabe flow for arbitrary initial energy

TL;DR

This paper proves the full convergence of the fractional Yamabe flow from any initial energy level, assuming the fractional positive mass conjecture holds, extending previous results limited to small initial energies.

Contribution

It establishes the convergence of the fractional Yamabe flow for all initial energies under the fractional positive mass conjecture, removing previous energy restrictions.

Findings

Full convergence result for arbitrary initial energy

Dependence on the fractional positive mass conjecture

Extension of previous small-energy convergence results

Abstract

Since the seminal paper of Graham and Zworski (Invent. Math. 2003), conformal geometric problems are studied in the fractional setting. We consider the convergence of fractional Yamabe flow, which is previously known under small initial energy assumption. Inspired by the deep work of Brendle (J. Diff. Geom. 2005), we obtain the full convergence result for arbitrary initial energy, whenever the (fractional) positive mass conjecture is valid.