Classification of 3-dimensional complete rectifiable steady and expanding gradient Ricci solitons

TL;DR

This paper classifies 3D complete rectifiable steady and expanding gradient Ricci solitons, showing they are either quotients of Euclidean space, Bryant solitons, or rotationally symmetric, with implications for positive Ricci curvature cases.

Contribution

It provides a complete classification of 3D rectifiable steady and expanding gradient Ricci solitons, identifying their geometric structures and symmetry properties.

Findings

Steady solitons are either quotients of ^3 or Bryant solitons.

Any steady soliton with positive Ricci curvature is isometric to Bryant.

Expanding solitons with positive Ricci curvature are rotationally symmetric.

Abstract

Let $(M,g,f)$ be a 3-dimensional complete steady gradient Ricci soliton. Assume that $M$ is rectifiable, that is, the potential function can be written as $f=f(r)$, where $r$ is a distance function. Then, we prove that $M$ is isometric to (1) a quotient of $\mathbb{R}^3$, or (2) the Bryant soliton. In particular, we show that any 3-dimensional complete rectifiable steady gradient Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. Furthermore, we show that any $3$-dimensional complete rectifiable expanding gradient Ricci soliton with positive Ricci curvature is rotationally symmetric.