Ricci almost solitons

TL;DR

This paper introduces Ricci almost solitons as an extension of gradient Ricci solitons, providing foundational results, curvature estimates, and topological insights within the framework of weighted manifold theory.

Contribution

It defines Ricci almost solitons, establishes existence and rigidity results, and explores their geometric and topological properties using advanced weighted manifold techniques.

Findings

Existence and rigidity theorems for Ricci almost solitons

A-priori curvature estimates and isolation phenomena

Topological properties and differential identities derived

Abstract

We introduce a natural extension of the concept of gradient Ricci soliton: the Ricci almost soliton. We provide existence and rigidity results, we deduce a-priori curvature estimates and isolation phenomena, and we investigate some topological properties. A number of differential identities involving the relevant geometric quantities are derived. Some basic tools from the weighted manifold theory such as general weighted volume comparisons and maximum principles at infinity for diffusion operators are discussed.