Geometric and analytical results for $\rho$-Einstein solitons

TL;DR

This paper investigates the geometric and analytical properties of complete noncompact $ ho$-Einstein solitons, including spectral analysis, volume growth estimates, and rigidity conditions, contributing new insights into their structure under Ricci-Bourguignon flow.

Contribution

It provides new spectral, volume growth, and rigidity results for $ ho$-Einstein solitons, extending classical geometric analysis to this broader class of self-similar solutions.

Findings

Spectrum analysis of drifted Laplacian for gradient shrinking solitons

New volume growth estimates for geodesic balls

Discussion of rigidity cases for $ ho$-Einstein solitons

Abstract

In this article, we study geometric and analytical features of complete noncompact $\rho$-Einstein solitons, which are self-similar solutions of the Ricci-Bourguignon flow. We study the spectrum of the drifted Laplacian operator for complete gradient shrinking $\rho$-Einstein solitons. Moreover, similar to classical results due to Calabi--Yau and Bishop for complete Riemannian manifolds with nonnegative Ricci curvature, we prove new volume growth estimates for geodesic balls of complete noncompact $\rho$-Einstein solitons. In particular, the rigidity case is discussed. In addition, we establish weighted volume growth estimates for geodesic balls of such manifolds.