Analytic and geometric properties of generic Ricci solitons

TL;DR

This paper classifies three-dimensional generic shrinking Ricci solitons, showing they are quotients of standard spaces, and introduces analytical tools like the Omori-Yau maximum principle for the $X$-Laplacian applicable in broader contexts.

Contribution

It provides classification results for 3D generic shrinking Ricci solitons and establishes the Omori-Yau maximum principle for the $X$-Laplacian without restrictions on $X$.

Findings

Classifies 3D generic shrinking Ricci solitons as quotients of standard spaces.

Proves the Omori-Yau maximum principle for the $X$-Laplacian on all generic Ricci solitons.

Introduces analytical tools useful for further geometric analysis.

Abstract

The aim of this paper is to prove some classification results for generic shrinking Ricci solitons. In particular, we show that every three dimensional generic shrinking Ricci soliton is given by quotients of either $\mathds{S}^3$, $\erre\times\mathds{S}^2$ or $\erre^3$, under some very weak conditions on the vector field $X$ generating the soliton structure. In doing so we introduce analytical tools that could be useful in other settings; for instance we prove that the Omori-Yau maximum principle holds for the $X$-Laplacian on every generic Ricci soliton, without any assumption on $X$.