Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres

TL;DR

This paper investigates the existence and uniqueness of constant scalar curvature metrics close to homogeneous metrics on spheres, revealing bifurcation points and rigidity phenomena through variational methods.

Contribution

It characterizes all critical points of the Hilbert-Einstein functional near homogeneous metrics on spheres, identifying bifurcation and local rigidity phenomena for specific symmetry groups.

Findings

Identification of bifurcation points for homogeneous metrics

Demonstration of local rigidity near certain metrics

Characterization of critical points of the Hilbert-Einstein functional

Abstract

We study existence and non-existence of constant scalar curvature metrics conformal and arbitrarily close to homogeneous metrics on spheres, using variational techniques. This describes all critical points of the Hilbert-Einstein functional on such conformal classes, near homogeneous metrics. Both bifurcation and local rigidity type phenomena are obtained for 1-parameter families of U(n+1), Sp(n+1) and Spin(9)-homogeneous metrics.