Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions

TL;DR

This paper proves the existence of singular rotationally symmetric steady and expanding gradient Ricci solitons in higher dimensions using fixed point methods, providing new solutions and analyzing their asymptotic behavior near the origin.

Contribution

It introduces a fixed point approach to establish the existence of infinitely many singular Ricci solitons in higher dimensions with detailed asymptotic analysis.

Findings

Existence of infinitely many solutions for the Ricci soliton equation in higher dimensions.

Characterization of the asymptotic behavior of solutions near the origin.

Conditions for the uniqueness of global singular solutions.

Abstract

By using fixed point argument we give a proof for the existence of singular rotationally symmetric steady and expanding gradient Ricci solitons in higher dimensions with metric $g=\frac{da^2}{h(a^2)}+a^2g_{S^n}$ for some function $h$ where $g_{S^n}$ is the standard metric on the unit sphere $S^n$ in $\mathbb{R}^n$ for any $n\ge 2$. More precisely for any $\lambda\ge 0$ and $c_0>0$, we prove that there exist infinitely many solutions $h\in C^2((0,\infty);\mathbb{R}^+)$ for the equation $2r^2h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+rh_r(r)(rh_r(r)-\lambda r-(n-1))$, $h(r)>0$, in $(0,\infty)$ satisfying $\underset{\substack{r\to 0}}{\lim}\,r^{\sqrt{n}-1}h(r)=c_0$ and prove the higher order asymptotic behaviour of the global singular solutions near the origin. We also find conditions for the existence of unique global singular solution of such equation in terms of its asymptotic behaviour near the origin.