Ground state of the conformal flow on $\mathbb{S}^3$

TL;DR

This paper analyzes the conformal flow on the 3-sphere, proving the ground state family maximizes energy under constraints and establishing their nonlinear orbital stability, with spectral analysis revealing a supersymmetric structure.

Contribution

It identifies the ground state family as energy maximizers and proves their nonlinear orbital stability using spectral properties and handling degeneracy issues.

Findings

Ground state family maximizes energy constrained to fixed charge.

Spectral analysis reveals supersymmetric structure of linearized flow.

Proved nonlinear orbital stability of the ground state family.

Abstract

We consider the conformal flow model derived by Bizo\'n, Craps, Evnin, Hunik, Luyten, and Maliborski [Commun. Math. Phys. 353 (2017) 1179-1199] as a normal form for the conformally invariant cubic wave equation on $\mathbb{S}^3$. We prove that the energy attains a global constrained maximum at a family of particular stationary solutions which we call the ground state family. Using this fact and spectral properties of the linearized flow (which are interesting on their own due to a supersymmetric structure) we prove nonlinear orbital stability of the ground state family. The main difficulty in the proof is due to the degeneracy of the ground state family as a constrained maximizer of the energy.