Multiplicity of Solutions and Degenerate Solutions of Yamabe-Type Equations on Projective Spaces

TL;DR

This paper investigates the existence, multiplicity, and degeneracy of solutions to Yamabe-type equations on complex and quaternionic projective spaces, focusing on solutions invariant under specific symmetry groups.

Contribution

It demonstrates the existence of multiple and degenerate solutions to Yamabe-type equations on projective spaces with symmetry invariance.

Findings

Existence of multiple solutions invariant under symmetry groups.

Existence of degenerate solutions with symmetry invariance.

Solutions are studied on complex and quaternionic projective spaces.

Abstract

We consider Yamabe-type equations on Projective Spaces $\mathbb{C} {\bf P}^n$ and $\mathbb{H} {\bf P}^n$ with the respectives canonical metrics, and study the existence and multiplicity of solutions of Yamabe-type equation, which are invariant by cohomogeneity one actions of $ U(n)$ and $Sp(n)$ respectively. We also prove the existence of degenerate solutions of the Yamabe-type equationn on $\mathbb{C} {\bf P}^n$ and $\mathbb{H} {\bf P}^n$, which are invariant by cohomogeneity one actions of $ U(n)$ and $Sp(n)$ respectively.