Invariant Solutions of the Yamabe equation on the Koiso-Cao Soliton

TL;DR

This paper investigates the Yamabe equation on a specific Ricci soliton metric on a complex surface, proving the uniqueness of a certain invariant solution under symmetry constraints.

Contribution

It establishes the existence and uniqueness of a $U(2)$-invariant solution to the Yamabe equation on the Koiso-Cao Ricci soliton.

Findings

Exactly one $U(2)$-invariant solution exists up to homothety.

The solution is unique among $U(2)$-invariant solutions.

The result enhances understanding of geometric PDEs on Ricci solitons.

Abstract

We consider the non-trivial Ricci soliton on $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$ constructed by Koiso and Cao. It is a K\"ahler metric invariant by the $U(2)$ action on $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$. We study its Yamabe equation and prove it has exactly one $U(2)-$invariant solution up to homothecies.