Bifurcation and Local Rigidity of Homogeneous Solutions to the Yamabe Problem on Aloff-Wallach Spaces

TL;DR

This paper investigates the bifurcation points and local rigidity of homogeneous solutions to the Yamabe problem on Aloff-Wallach spaces by analyzing Morse index changes in parameterized solutions.

Contribution

It constructs parameter families of solutions and identifies bifurcation and rigidity points, advancing understanding of solution structure on Aloff-Wallach spaces.

Findings

Identification of bifurcation points through Morse index analysis

Demonstration of local rigidity at certain solutions

Construction of solution families on Aloff-Wallach spaces

Abstract

We construct 1-parameter families of well-known solutions to the Yamabe problem defined on Aloff-Wallach Spaces to determine bifurcation instants for these homogeneous spaces by examining changes in the Morse index of these metrics as the parameter varies over the positive real numbers. A bifurcation point for such families is an accumulation point of other solutions to the Yamabe problem, while a local rigidity point is an isolated solution of this problem, i.e., it is not a bifurcation point.