Rotational symmetry of non negatively curved expanding gradient Ricci solitons

TL;DR

This paper proves that certain non-negatively curved expanding gradient Ricci solitons with smooth convergence to a specific cone are rotationally symmetric, extending the Perelman conjecture to the expanding case.

Contribution

It establishes rotational symmetry for expanding gradient Ricci solitons with nonnegative curvature and smooth asymptotic cone convergence, generalizing known results for the Bryant soliton.

Findings

Proves rotational symmetry under specified conditions

Extends the Perelman conjecture to expanding solitons

Builds on Brendle's and Chodosh's recent work

Abstract

Let $(M^n,g,\nabla f)$, $n\geq 3$, be an expanding gradient Ricci soliton with nonnegative sectional curvature whose asymptotic cone is isometric to $C(\mathbb{S}^{n-1}(c))$ where $\mathbb{S}^{n-1}(c)$ is the standard $(n-1)$-sphere of curvature $1/c^2$, with $c\in(0,1)$. We prove that if the convergence to the asymptotic cone is smooth, $(M^n,g,\nabla f)$ is rotationally symmetric. This is the expanding analogue of the Perelman conjecture on the Bryant soliton and this work is based on the proof by Brendle \cite{Bre-Rot-3d}. This has also been proved recently by Chodosh \cite{Cho-EGS}.