A new proof for the existence of rotationally symmetric gradient Ricci solitons

TL;DR

This paper presents a new proof for the existence of rotationally symmetric steady and expanding gradient Ricci solitons in dimensions 2 to 4, providing explicit solutions and analyzing their asymptotic behavior.

Contribution

The authors develop a novel proof technique for the existence of specific Ricci solitons with rotational symmetry, including explicit solutions and asymptotic analysis for dimensions 2 to 4.

Findings

Existence of unique solutions for the Ricci soliton equation in specified dimensions.

Explicit construction of solutions with given initial conditions.

Analysis of the asymptotic behavior of solutions for various parameters.

Abstract

We give a new proof for the existence of rotationally symmetric steady and expanding gradient Ricci solitons in dimension $n+1$, $2\le n\le 4$, with metric $g=\frac{da^2}{h(a^2)}+a^2d\,\sigma$ for some function $h$ where $d\sigma$ is the standard metric on the unit sphere $S^n$ in $\mathbb{R}^n$. More precisely for any $\lambda\ge 0$, $2\le n\le 4$ and $\mu_1\in\mathbb{R}$, we prove the existence of unique solution $h\in C^2((0,\infty))\cap C^1([0,\infty))$ 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 $h(0)=1$, $h_r(0)=\mu_1$. We also prove the existence of unique analytic solution of the about equation on $[0,\infty)$ for any $\lambda\ge 0$, $n\ge 2$ and $\mu_1\in\mathbb{R}$. Moreover we will prove the asymptotic behaviour of the solution $h$ for any $n\ge 2$, $\lambda\ge 0$ and $\mu_1\in\mathbb{R}\setminus\{0\}$.