Geometric and analytic results for Einstein solitons

TL;DR

This paper investigates properties of Einstein solitons, providing bounds, asymptotic behavior, and examples, thereby advancing understanding of their geometric and analytic structure.

Contribution

It introduces new bounds, asymptotic analysis, and explicit examples for gradient Einstein solitons, enriching the theoretical framework and construction methods.

Findings

Lower bound for scalar curvature under potential function assumptions

Finiteness of fundamental group and weighted volume for noncompact solitons

Explicit examples of Einstein solitons as warped metrics

Abstract

We compute a lower bound for the scalar curvature of a gradient Einstein soliton under a certain assumption on its potential function. We establish an asymptotic behavior of the potential function on a noncompact gradient shrinking Einstein soliton. As a result, we obtain the finiteness of its fundamental group and its weighted volume. We also prove some geometric and analytic results for constructing gradient Einstein solitons that are realized as warped metrics, and we give a few explicit examples.