On a classification of the gradient shrinking solitons

TL;DR

This paper offers an alternative proof of Perelman's classification of gradient shrinking solitons, extends results in three dimensions by removing non-collapsing assumptions, and classifies high-dimensional solitons with zero Weyl curvature.

Contribution

It provides a new proof of known results, generalizes classification in three dimensions, and classifies certain high-dimensional solitons with specific curvature conditions.

Findings

Alternative proof of Perelman's result

Removal of non-collapsing assumption in 3D

Classification of high-dimensional solitons with vanishing Weyl tensor

Abstract

The main purpose of this article is to provide an alternate proof to a result of Perelman on gradient shrinking solitons. In dimension three we also generalize the result by removing the $\kappa$-non-collapsing assumption. In high dimension this new method allows us to prove a classification result on gradient shrinking solitons with vanishing Weyl curvature tensor, which includes the rotationally symmetric ones.