Bifurcation for the constant scalar curvature equation and harmonic Riemannian submersions

TL;DR

This paper investigates bifurcation phenomena for the constant scalar curvature equation on Riemannian manifolds with harmonic submersions, providing existence criteria, characterizations of degeneracy points, and symmetry considerations.

Contribution

It introduces new bifurcation criteria and characterizations for constant scalar curvature metrics in the context of harmonic Riemannian submersions, including specific results for quaternionic Hopf fibrations.

Findings

Existence of bifurcation points under certain conditions.

Characterization of degeneracy points along the family of metrics.

Symmetry-breaking bifurcation only at the round metric in quaternionic Hopf fibrations.

Abstract

We study bifurcation for the constant scalar curvature equation along a one-parameter family of Riemannian metrics on the total space of a harmonic Riemannian submersion. We provide an existence theorem for bifurcation points and a criterion to see that the conformal factors corresponding to the bifurcated metrics must be indeed constant along the fibers. In the case of the canonical variation of a Riemannian submersion with totally geodesic fibers, we characterize discreteness of the set of all degeneracy points along the family and give a sufficient condition to guarantee that bifurcation necessarily occurs at every point where the linearized equation has a nontrivial solution. In the model case of quaternionic Hopf fibrations, we show that symmetry-breaking bifurcation does not occur except at the round metric.